Mapping Diagrams to Visualize Complex Analysis

Martin Flashman, Dec 9, 2015

Mapping Diagrams to Visualize Complex Analysis illustrates the use of mapping diagrams in the analysis of functions of a complex variable. This work was motivated initially by [url]https://sites.google.com/site/visualcomplexanalysis/[/url] Workshop material prepared for IV Nordic GeoGebra Conference, 20.-22. September, Copenhagen, Denmark By Rokas Tamošiūnas (rokas.tamosiunas@ksu.lt) I have organized an MAA mini-course on this topic: Joint Mathematics Meetings Denver, CO Jan. 15&17, 2020. MAA Minicourse 2020: Visual Complex Analysis- GeoGebra Tools and Mapping Diagrams GeoGebra Book. [url]https://www.geogebra.org/m/rsqxtq9t[/url] A guide on how to explore complex integration in other ways with GeoGebra can be found at: [url]http://goo.gl/1aToaT[/url] Another publication with a little bit different approach can be found at: [url]http://ggijro1.files.wordpress.com/2013/10/visualizing_complex_integral_final1.pdf [/url].

Table of Contents

  1. Introduction to Complex Numbers and Arithmetic
    1. Complex Plane
    2. Complex Addition
    3. Complex Multiplication
    4. Complex 1/z
    5. Experimenting with Complex Arithmetic
  2. Linear and Quadratic Functions and Equations
    1. Table and Mapping Diagram for A Complex Linear Function
    2. Complex Linear Mapping Diagram II: f(z) = Az
    3. Complex Linear Mapping Diagram III (lines)
    4. Complex Linear Mapping Diagrams I (Circles)
    5. Complex Mapping Diagrams for Quadratic Function and Solving Real Quadratic Equations
    6. Mapping Diagrams for Complex Quadratic Functions & Equations
    7. Mapping Diagram Solving A Quadratic Equation(Real and Complex)
    8. Complex Variable Function Mapping Diagrams and Solving Real Cubic Equations
  3. Mapping Diagrams to Visualize Elementary Functions
    1. Mapping Diagrams: Complex Analysis -functions with complex parameters. circles
    2. Mapping Diagrams: Complex Analysis-functions with lines or circles
    3. Mapping Diagram: Complex Analysis. Realign Target.
    4. Complex Linear Mapping Diagram f(z) = Bz; parametric curve
    5. Moebius Transformation: MD circle with spherical projection
    6. Complex Mapping Diagram for Moebius Function (circles)
    7. Mapping diagram for Moebius Function
    8. Mapping diagram for Moebius Transformation: roots and poles, lines and circles
    9. Mapping Diagram for Complex Power Function (Circles)
    10. Mapping Diagram for z^n with Curve on Riemann Surface
    11. Mapping Diagram for z^n
    12. Complex Mapping diagram with homotopy
    13. Mapping Diagram Powers Homotopy
    14. Mapping Diagrams for Quartic Equations
    15. Mapping Diagrams for Visualizing Roots for Polynomials
    16. Complex plane transformations: mapping a line
    17. Complex square mapping between two planes
  4. Limits and Continuity for Complex Functions
    1. Mapping Diagrams: Complex Analysis- functions and continuity
    2. Limits for Complex Functions
    3. Mapping Diagram:Limits for Complex Functions
    4. Limits for Complex Functions
    5. Limits for Complex Functions
    6. Continuity and Open Sets (Work in Progress)
  5. Mapping Diagrams to Visualize Differentiation and Applications
    1. Mapping Diagram Visualizing Complex Derivative:Powers
    2. Mapping Diagram Visualizing Complex Differential
    3. Mapping Diagram for Newton's Method in Complex Analysis
  6. Mapping Diagrams to Visualize Integration
    1. Complex Line Integral: Definition and F.T. Examples
    2. Mapping Diagram for Cauchy Integral Formula
    3. Complex Integration from list
    4. Integral Examples from List
    5. Mapping Diagram: Complex Integration Parametric Function
    6. Mapping Diagram for Complex Integration over A Circle
    7. Mapping Diagram: Complex Integration of 1/z
    8. Complex Integration: z^2 over line segment
    9. Complex Integration: z^2 over curve
    10. The Winding Number of a Curve about a Point
  7. Series: Taylor and Laurent
    1. Taylor Polynomials for Complex Functions: Mapping Diagrams

1.

Introduction to Complex Numbers and Arithmetic

  • 1. Complex Plane
  • 2. Complex Addition
  • 3. Complex Multiplication
  • 4. Complex 1/z
  • 5. Experimenting with Complex Arithmetic

1.1.

Complex Plane

[b][u]Complex Numbers   [/u]  [math]\mathbb{C}[/math][/b][b][br][br]Definition: [u]The "number" [math]i[/math][/u][/b][b][u]:[br][/u][br][/b]To solve the equation [math]x^2+1=0[/math] we create a symbol [math]i[/math] and declare [br]that [math]i^2=−1[/math] so [math]i[/math] solves the equation  and we write [math]i=\sqrt{-1}[/math].[br][br][b]Definitions: [u]A complex number[/u] [math]z[/math][/b][b] is a number that can be expressed in the form [math]z=a+bi[/math][/b][b] where [/b][math]a[/math][b] and [math]b[/math][/b][b] are real numbers. [br][br]If [/b][math]z=a+bi[/math][b] then [math]a[/math][/b][b] is called the real part of [math]z[/math][/b][b] and [math]bi[/math][/b][b] (and sometimes [math]b[/math][/b][b]) is called the imaginary part of [/b][math]z[/math][b].[/b][br][br][b]Examples: [math]3+2i[/math][/b][b] and [math]−2−i[/math][/b][b].[br][br][u][b]Complex Geometry- The Complex Plane[/b][/u][br][br][/b]Complex numbers are identified with points in a cartesian plane by having [math]a+bi[/math] identified with the point with coordinates [math](a,b)[/math] or with position vectors by identifying [br][math]a+bi[/math] with the vector [math]\left(\begin{matrix}a\\b\end{matrix}\right)[/math].[br]With this identification [math]i[/math] is identified with the point [math](0,1)[/math] or the vector [math]\left(\begin{matrix}0\\1\end{matrix}\right)[/math] and [math]−i[/math] is identified with [math](0,−1)[/math] or the vector [math]\left(\begin{matrix}0\\-1\end{matrix}\right)[/math].[b][b][br][br] [center][img width=269,height=224]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA+4AAAM+CAIAAAALuMzGAAAgAElEQVR4nOzdcaie5Z0n/BuOIZuGNBVcCSKdjBCWImJLISuOdEQGRERKV4VpKW6mFME6Q3emiEORJmRlVra0cZDBKc3WUIqIsNgZCVOx6Hlb14R5ZQ9BnEPHLce3cfM6jmNeR9pszJxe7x/nzDFNniTnOec53/u6r+fzIQwzmVzPuc83v+vON49P7qsr63b8+PG5ubnjx4+v/6W4uJMnTy4sLJw8ebLvC2mfqGNETXtMdYyoY0QdM27U3fq/5Nzc3Ozs7Nzc3PpfiotbWFiYnZ1dWFjo+0LaJ+oYUSe98cYbfV/CVDDVMaKOEXXMuFGr8kNiI8WIOkbUMV/5ylcuu+yy+++/v+8LaZ+pjhF1jKhjVPmW2Ugxoo4Rdcbf/M3fzMzMdF03MzPzox/9qO/LaZypjhF1jKhjVPmW2Ugxoo4RdcB3v/vdyy67rOu6j370o13XXXbZZQcPHuz7olpmqmNEHSPqGFW+ZTZSjKhjRL2hfv3rXz/44INd13Vdd9NNN638z67rHnzwwV//+td9X2CbTHWMqGNEHaPKt8xGihF1jKg3zq9+9au77rqr67pNmzYdPHhw7969Xdft27fv4MGDmzZt6rrurrvu+tWvftX3ZTbIVMeIOkbUMap8y2ykGFHHiHqDfPDBBzfccEPXdR/72Md+/OMfl1JWqnwp5cc//vH27du7rrvhhhs++OCDvi+2NaY6RtQxoo5R5VtmI8WIOkbUG2fv3r2//du//Xd/93cr/+dKlS+lvPbaazt37lz5P5kgUx0j6hhRx6jyLbORYkQdI+oNdfYhI+dU+XP+v0yQqY4RdYyoY3qo8k57jXHWWoyoY0Qdc36VZ4OY6hhRx4g6pofTXgGonyoP0B5VHmAqqPIA7VHlAaaCKg/QHlUeYCqo8gDtUeUBpoIqD9AeVR5gKqjyAO1R5QGmgioP0B7PlR8ST3WNEXWMqGNU+RhTHSPqGFHH9PBceae9xjhrLUbUMaKOUeVjTHWMqGNEHdPDaa+qfIyNFCPqGFHHqPIxpjpG1DGijlHlW2YjxYg6RtQxqnyMqY4RdYyoY1T5ltlIMaKOEXWMKh9jqmNEHSPqGFW+ZTZSjKhjRB2jyseY6hhRx4g6RpVvmY0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjVPmW2Ugxoo4RdYwqH2OqY0QdI+oYVb5lNlKMqGNEHaPKx5jqGFHHiDqmhyrvtNcYZ63FiDpG1DGqfIypjhF1jKhjejjtFYD6qfIA7VHlAaaCKg/QHlUeYCqo8gDtUeUBpoIqD9AeVR5gKqjyAO1R5QGmgioP0B5VHmAqqPIA7fFc+SHxVNcYUceIOkaVjzHVMaKOEXVMD8+Vd9prjLPWYkQdI+oYVT7GVMeIOkbUMT2c9qrKx9hIMaKOEXWMKh9jqmNEHSPqGFW+ZTZSjKhjRB2jyseY6hhRx4g6RpVvmY0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjxq7yS8+fuZALPZfm7FU//elPZ2dnf/rTn461am1fa8pXLf3uHjlypNorbGbV+RuptitsZtWRI0eWprraKxziqhdffHHvWb773e8+88wzTzzxxKOPPvrDH/6whitse9XKDaTaK2xm1bFjx0aWnnqusJlVK/fqaq+wmVVLU32hsnf+qm5ubm72wuYu8F67Vb2sWvrjoeYrbGbV+VW+tiu0yqqLrFp6A37Fo48+WtsVtr1q5QZS7RU2s2qpX55f5eu5QqusGnfV0lSvfpV35Ye0yrvysVXelY+t8q78RqyanZ3dd5aDBw/+8Ic/PHTokHflM6u8Kx9b5V352CrvysdWjf2u/MhXGcucz8qn+KRajKhjRB3js/IxpjpG1DGijunhn70ed9prirPWYkQdI+oYVT7GVMeIOkbUMT2c9gpA/VR5gPao8gBTQZUHaI8qDzAVVHmA9qjyAFNBlQdojyoPMBVUeYD2qPIAU0GVB2iPKg8wFVR5gPZ4rvyQeKprjKhjRB2jyseY6hhRx4g6pofnyjvtNcZZazGijhF1jCofY6pjRB0j6pgeTntV5WNspBhRx4g6RpWPMdUxoo4RdYwq3zIbKUbUMaKOUeVjTHWMqGNEHaPKt8xGihF1jKhjVPkYUx0j6hhRx6jyLbORYkQdI+oYVT7GVMeIOkbUMap8y2ykGFHHiDpGlY8x1TGijhF1jCrfMhspRtQxoo5R5WNMdYyoY0Qdo8q3zEaKEXWMqGNU+RhTHSPqGFHHqPIts5FiRB0j6hhVPsZUx4g6RtQxPVR5p73GOGstRtQxoo5R5WNMdYyoY0Qd08NprwDUT5UHaI8qDzAVVHmA9qjyAFNBlQdojyoPMBVUeYD2qPIAU0GVB2iPKg8wFVR5gPao8gBTQZUHaI/nyg+Jp7rGiDpG1DGqfIypjhF1jKhjeniuvNNeY5y1FiPqGFHHqPIxpjpG1DGijunhtFdVPsZGihF1jKhjVPkYUx0j6hhRx6jyLbORYkQdI+oYVT7GVMeIOkbUMap8y2ykGFHHiDpGlY8x1TGijhF1jCrfMhspRtQxoo5R5WNMdYyoY0Qdo8q3zEaKEXWMqGNU+RhTHSPqGFHHqPIts5FiRB0j6hhVPsZUx4g6RtQxqnzLbKQYUceIOkaVjzHVMaKOEXWMKt8yGylG1DGijlHlY0x1jKhjRB3TQ5V32muMs9ZiRB0j6hhVPsZUx4g6RtQxPZz2CkD9VHmA9qjyAFNBlQdojyoPMBVUeYD2qPIAU0GVB2iPKg8wFVR5gPao8gBTQZUHaI8qDzAVVHmA9niu/JB4qmuMqGNEHaPKx5jqGFHHiDqmh+fKO+01xllrMaKOEXWMKh9jqmNEHSPqmB5Oe1XlY2ykGFHHiDpGlY8x1TGijhF1jCrfMhspRtQxoo5R5WNMdYyoY0Qdo8q3zEaKEXWMqGNU+RhTHSPqGFHHqPIts5FiRB0j6hhVPsZUx4g6RtQxqnzLbKQYUceIOkaVjzHVMaKOEXWMKt8yGylG1DGijlHlY0x1jKhjRB2jyrfMRooRdYyoY1T5GFMdI+oYUceo8i2zkWJEHSPqGFU+xlTHiDpG1DE9VHmnvcY4ay1G1DGijlHlY0x1jKhjRB3Tw2mvANRPlQdojyoPMBVUeYD2qPIAU0GVB2iPKg8wFVR5gPao8gBTQZUHaI8qDzAVVHmA9qjyAFNBlQdoj+fKD4mnusaIOkbUMap8jKmOEXWMqGN6eK68015jnLUWI+oYUceo8jGmOkbUMaKO6eG0V1U+xkaKEXWMqGNU+RhTHSPqGFHHqPIts5FiRB0j6hhVPsZUx4g6RtQxqnzLbKQYUceIOkaVjzHVMaKOEXWMKt8yGylG1DGijlHlY0x1jKhjRB2jyrfMRooRdYyoY1T5GFMdI+oYUceo8i2zkWJEHSPqGFU+xlTHiDpG1DGqfMtspBhRx4g6RpWPMdUxoo4RdYwq3zIbKUbUMaKOUeVjTHWMqGNEHdNDlXfaa4yz1mJEHSPqGFU+xlTHiDpG1DE9nPYKQP1UeYD2qPIAU0GVB2iPKg8wFVR5gPao8gBTQZUHaI8qDzAVVHmA9qjyAFNBlQdojyoPMBVUeYD2eK78kHiqa4yoY0Qdo8rHmOoYUceIOqaH58o77TXGWWsxoo4RdYwqH2OqY0QdI+qYHk57VeVjbKQYUceIOkaVjzHVMaKOEXWMKt8yGylG1DGijlHlY0x1jKhjRB2jyrfMRooRdYyoY1T5GFMdI+oYUceo8i2zkWJEHSPqGFU+xlTHiDpG1DGqfMtspBhRx4g6RpWPMdUxoo4RdYwq3zIbKUbUMaKOUeVjTHWMqGNEHaPKt8xGihF1jKhjVPkYUx0j6hhRx6jyLbORYkQdI+oYVT7GVMeIOmY1Uf/FX5Tdu8uWLWXLlvLpT5cDB8riYujyZmfLnXeWbdvKpk1ly5Zyyy3lySd/4xe8+27ZtKl0XXn44dAlrVkPVd5przHOWosRdYyoY1T5GFMdI+qYi0e9uFhuu6103bk/PvOZcurUhl/bww+P+NJdV+6558Nf89hjw+jxpZfTXgGonyoPbJBvfrN0Xdm5sxw+XBYXy+Ji+clPyjXXlK4rDz20sV/6pZdK15XNm8sjj5R33imllPfeK9/85vJ78I8/vvzLdu8eRo9fA1UeYCqo8sAG2bWrdF157bXf+MlXXy1dV66+eryXWnpDffXuvLN0XfnOd879+YMHS9eV668f76sPkSoPMBVUeSBs6f3ycZeMVeW3bStdV86cOffnFxfP/erjvvJQtPg9AXAeVR5Ievnl0nXlxhvHWzWpwn3mTOm6sm3b5F+5Ni1+TwCcR5UHMk6dKgcPliuvLJs2lZ/8ZLy1kyrcTz5Zuq588YuTf+XatPg9AXAeVR4I+Nznlkvzli3l5ZfHXj6Rwv3+++Waa8q2beVnP5vwK1eoxe8JgPOo8kDA7t3l9tuXH19zxRXl+9+/xK8f+RzJc36MZXGx3HprufLKc/8iocpfkOfKx3iAboyoY0Qdo8rHmOoYUcesIeqXXipXXlm6rjz77MV+2cSr/N13l507y4kTo79Q/Xp4rrzTXmMcaxcj6hhRx6jyMaY6RtQxa4v6uedK15Xdu8dYsp7CfeZM+dznytat5fXXJ/zKST2c9qrKx7hnxYg6RtQxqnyMqY4Rdczaol56IuSmTWMsWXPhfuedcuutpevKt7414VcOU+Vb5p4VI+oYUceo8jGmOkbUMRePevv20nXlzTfP/fnTp899IuQlra1wv/xyufrq0nXlkUcm/Mp5qnzL3LNiRB0j6hhVPsZUx4g65uJRf/GLpevK+XeXxx4rXVc++9kxvtAaCvfhw2Xz5rJ9ezl8eMKv3AtVvmXuWTGijhF1jCofY6pjRB1z8aiPHVt++uShQ8sHr/7yl+Xxx8u2bWVmprzyyhhfaNzCPT9ftmwpV15Z5ucn/Mp9UeVb5p4VI+oYUceo8jGmOkbUMZeM+uDBEU+emZkpTz65sRe28hj7Sz79RpW/IFU+xj0rRtQxoo5R5WNMdYyoY1YT9d/+bfnc55bfid+xo3zpS+W11zb8wjZvVuXXTZWPcc+KEXWMqGNU+RhTHSPqGFHHqPIts5FiRB0j6hhVPsZUx4g6RtQxPVR5p73GONYuRtQxoo5R5WNMdYyoY0Qd08NprwDUT5UHaI8qDzAVVHmA9qjyAFNBlQfW7PwHwlAJvycAU0GVB9ZMla+W3xOAqaDKA2umylfL7wnAVFDlgTVT5avl9wRgKqjywJqp8tXyXPkh8VTXGFHHiDpGlY8x1TGijlHlY8Z+rvzSggu50AudverIkSOzs7NHjhwZa9XavtaUr1pYWFg6WLfaK2xm1cJ5Z63VdoXNrFo5LrraKxziqhdffHHvWQ4cOPDEE0/s3bt3z549hw4dquEK21618K83kGqvsJlV8/PzI8/FrOcKB7pqpbiv5seAvq9BrFqa6guVvfNXdQsLC7MXtnCBY2Ot6mXVxZfUcIXNrFo4r8rXdoVWWXWRVUtvwK/Ys2dPbVfY9qqVG0i1V9jMqqX3As5fW88VDnTVmqt85d/XIFYtTfXqV3lXfkirFrwrn1q14F351Crvym/EqtnZ2X1nefTRRw8dOrRv3z7vymdWLXhXPrXKu/IbtMq78j2uGvtd+ZGvMpaVP4nX/1Jc3MJ5/ZINIuoYUcf4rHyMqY4RdYzPyseMO9Wq/JC4Z8WIOkbUMap8jKmOEXWMKh+jyrfMPStG1DGijlHlY0x1jKhjVPkYVb5l7lkxoo4RdYwqH2OqY0Qdo8rHqPItc8+KEXWMqGNU+RhTHSPqGFU+RpVvmXtWjKhjRB2jyseY6hhRx6jyMT1Ueae9xpx0rF2KqGNEHaPKx5jqGFHHiDpm3Kj99QpgKqjyAO1R5QGmgioP0B5VHmAqqPIA7VHlAaaCKg/QHlUeYCqo8gDtUeUBpoIqD9AeVR5gKqjyAO3xXPkh8VTXGFHHiDpGlY8x1TGijhF1TA/PlXfaa4xj7WJEHSPqGFU+xlTHiDpG1DE9nPaqysfYSDGijhF1jCofY6pjRB0j6hhVvmU2UoyoY0Qdo8rHmOoYUceIOkaVb5mNFCPqGFHHqPIxpjpG1DGijlHlW2YjxYg6RtQxqnyMqY4RdYyoY1T5ltlIMaKOEXWMKh9jqmNEHSPqGFW+ZTZSjKhjRB2jyseY6hhRx4g6RpVvmY0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjeqjyTnuNcdZajKhjRB2jyseY6hhRx4g6pofTXgGonyoP0B5VHmAqqPIA7VHlAaaCKg/QHlUeYCqo8gDtUeUBpoIqD9AeVR5gKqjyAO1R5QGmgioP0B7PlR8ST3WNEXWMqGNU+RhTHSPqGFHH9PBceae9xjhrLUbUMaKOUeVjTHWMqGNEHdPDaa+qfIyNFCPqGFHHqPIxpjpG1DGijlHlW2YjxYg6RtQxqnyMqY4RdYyoY1T5ltlIMaKOEXWMKh9jqmNEHSPqGFW+ZTZSjKhjRB2jyseY6hhRx4g6RpVvmY0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjVPmW2Ugxoo4RdYwqH2OqY0QdI+oYVb5lNlKMqGNEHaPKx5jqGFHHiDqmhyrvtNcYZ63FiDpG1DGqfIypjhF1jKhjejjtFYD6qfIA7VHlAaaCKg/QHlUeYCqo8gDtUeUBpoIqD9AeVR5gKqjyAO1R5QGmgioP0B5VHmAqqPIA7fFc+SHxVNcYUceIOkaVjzHVMaKOEXVMD8+Vd9prjLPWYkQdI+oYVT7GVMeIOkbUMT2c9qrKx9hIMaKOEXWMKh9jqmNEHSPqGFW+ZTZSjKhjRB2jyseY6hhRx4g6RpVvmY0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjVPmW2Ugxoo4RdYwqH2OqY0QdI+oYVb5lNlKMqGNEHaPKx5jqGFHHiDpGlW+ZjRQj6hhRx6jyMaY6RtQxoo5R5VtmI8WIOkbUMap8jKmOEXWMqGN6qPJOe41x1lqMqGNEHaPKx5jqGFHHiDqmh9NeAaifKg/QHlUeYCqo8gDtUeUBpoIqD9AeVR5gKqjyAO1R5QGmgioP0B5VHmAqqPIA7VHlAaaCKg/QHs+VHxJPdY0RdYyoY1T5GFMdI+oYUcf08Fx5p73GOGstRtQxoo5R5WNMdYyoY0Qd08Npr6p8jI0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjVPmW2Ugxoo4RdYwqH2OqY0QdI+oYVb5lNlKMqGNEHaPKx5jqGFHHiDpGlW+ZjRQj6hhRx6jyMaY6RtQxoo5R5VtmI8WIOkbUMap8jKmOEXWMqGNU+ZbZSDGijhF1jCofY6pjRB0j6hhVvmU2UoyoY0Qdo8rHmOoYUceIOqaHKu+01xhnrcWIOkbUMap8jKmOEXWMqGN6OO0VgPqp8gDtUeUBpoIqD9AeVR5gKqjyAO1R5QGmgioP0B5VHmAqqPIA7VHlAaaCKg/QHlUeYCqo8gDt8Vz5IfFU1xhRx4g6RpWPMdUxoo4RdUwPz5V32muMs9ZiRB0j6hhVPsZUx4g6RtQxPZz2qsrH2Egxoo4RdYwqH2OqY0QdI+oYVb5lNlKMqGNEHaPKx5jqGFHHiDpGlW+ZjRQj6hhRx6jyMaY6RtQxoo5R5VtmI8WIOkbUMap8jKmOEXWMqGNU+ZbZSDGijhF1jCofY6pjRB0j6hhVvmU2UoyoY0Qdo8rHmOoYUceIOkaVb5mNFCPqGFHHqPIxpjpG1DGijlHlW2YjxYg6RtQxqnyMqY4RdYyoY3qo8k57jXHWWoyoY0Qdo8rHmOoYUceIOqaH014BqJ8qD9AeVR5gKqjyAO1R5QGmgioP0B5VHmAqqPIA7VHlAaaCKg/QHlUeYCqo8gDtUeUBpoIqvwa//GW5/fbyyCPllVf6//F//k/fcQD18Vz5IfFU1xhRx4g6RpVfg4ceKl1Xy4/XX+87jvq4gcSIOqaH58o77TXGWWsxoo4RdYwqP66f/7xs2lS2by9/8Afl3nvH+PH5z//zHXec+Pzn/3msVZf88Q//0Hci9XEDiRF1TA+nvaryMTZSjKhjRB2jyo/r7rtL15XvfGfshaY6RtQxoo5R5VtmI8WIOkbUMar8WI4eLV1XrrtuLWtNdYyoY0Qdo8q3zEaKEXWMqGMmXuUfeaTcdls5eHBSr1eXT32qdF154YW1rDXVMaKOEXWMKt8yGylG1DGijplslf/BD5b/LebmzRN5vbosfXd33LHG5aY6RtQxoo5R5VtmI8WIOkbUMZOt8g88sFzlf+/3JvJ6FTl1qlx1VZmZWfsTY9Y21e++W/74j8tTT63xi04nN5AYUceo8i2zkWJEHSPqmMlW+bffLvfdV772tfLuuxN5vYrs21e6rjzwwNpfYdypfv31sn9/ueKK0nXlnnvW/nWnkBtIjKhjVPmW2Ugxoo4RdczK48m5iBMnytat5fLL1/VXlNVM9fPPl89+ttx8c9m27TceHq/Kj8UNJEbUMap8y2ykGFHHiDpGlV+Ne+4pXVcOHFjXi6xmql94ofz+75d77ikHDpSjR8vmzar8WriBxIg6pocq77TXGGetxYg6RtQxqvwlzc2Vriuf+ERZXFzX66xhqlX5tXEDiRF1TA+nvQJQv4lU+cXF8vTT5f77y4ED5Y03JnRl1bjxxtJ15dlne/jSqjywNqo8wFRYf5V/++2ye3fZvXv5UyibN5dnnpnc9fXt6adL15Xbbuvnq6vywNqo8gBTYZ1V/vTp8ulPLz9q/Z13ll/qyitXu/zLXy63376uH9/61loue3GxvPTSqr67a64pMzPltdfW8lXWT5UH1qZb+kTOhVzokzpWWWWVVVZVu+rFF188+6Eol/yxmq/1J39y8iMf+fX//J//z8LCwoEDb1/ofKiRV/jzny/823/7L2Nd0vk/br117DRefbVcd13punL2cxlGrnrwwXe7rnzxi//c1+/y2VW+womyyiqrql3VLSwszF7YwgX+/axVVllllVXVrtq7d+/6q/w5X+uKK07/h//w5tL//u///T8tLfziFytN44UXZv/kT07OzIw4x+r8VX/91//jIx/5l4985F/++q//R1+/y2dX+UoytMoqqwaxyrvyVllllVWtrZqdnZ34u/J/+IcnX3rp+MLCwssvH1+pyD/5SY1pzM6++YlPnL7ppjNf+MKH3+PKpZ6/6vOf/+euKw8++G7sCs9f5V15q6yyam2rfFYeYCpM6mGU+/cvv84110zisjbAd75THn+8lFLeeqts3bp8tbt3j/7Fr722/L1c8gGUzz1Xvv/9Nf645MkrPisPrI3nyg/JSU91TRF1jKhjJlXlr7lm+XUefngSl7XBHnrow2/8r/5qxC+45ZbSdeW///dLvM6pU+v6oP/OnZd4fVV+bdxAYkQdM27UTnsdkgVnraWIOkbUMROp8i+8sPwiMzPlzTfHW3v69Np/nDmzxgt+991y+eXL13zddef+f599tnRd+cxnVvVSs7OrfQ/+29/+x69/ff7b3/7HlZ85duwSL67Kr40bSIyoY8aNWpUfEhspRtQxoo6ZSJVfeqJ815Xbb1/+mX37yle/eolVi4tlx451vavddeXmm9d4zQ8//OGLfP/7v3FV/+7fla67dM8e1xqmWpVfGzeQGFHHqPIts5FiRB0j6pj1V/n33iubNi2/yNNPl1LK6dNl+/byjW9ceu3996/3ufKPPbbGyz51qlx11fJl79r14Wfiv/Wt0nVlz541vuxFqPIxbiAxoo5R5VtmI8WIOkbUMeuv8ocPL7/Cpk3Ln3jZv79s317eemtS17hR/vzPP/z2l/5KsPTBm61by4kTk/9yqnyMG0iMqGNU+ZbZSDGijhF1zN69e7uu27dv35pf4a23ypYtpevKDTeUUsozz5SuK88+O7Er3DiLi2XnzuWuvGNHOXWqfPWrpevK/v0b8uVU+Rg3kBhRx6jyLbORYkQdI+qY9Vf5Uspzz5Wrry5dV664ouzYUQ4fntTVbbhDhz58Y/5LXyozM+XjHy+nT2/I11rNVJ858xv/Unblk0s33fThTz7//IZcXkvcQGJEHaPKt8xGihF1jKhjJlLll7z6ajl69NIPYq/Ntdf+xr+j/cEPNuoLrWaq33+/bN5ctm4t27aN/rF1a7ntto26wma4gcSIOkaVb5mNFCPqGFHHTLDKD9TSJ4IufmLURJjqGFHHiDpGlW+ZjRQj6hhRx6jypZTdu5er/CuvbOBXMdUxoo4RdUwPVd5przHOWosRdYyoY1T5UsrsbOm68oUvbOxXMdUxoo4RdUwPp70CUD9VfsmxY2s/OxagNqo8wFRQ5QHao8oDTAVVHqA9qjzAVFDlAdqjygNMBVUeoD2qPMBUUOUB2qPKA0wFVR6gPZ4rPySe6hoj6hhRx6jyMaY6RtQxoo7p4bnyTnuNcdZajKhjRB2jyseY6hhRx4g6pofTXlX5GBspRtQxoo5R5WNMdYyoY0Qdo8q3zEaKEXWMqGNU+RhTHSPqGFHHqPIts5FiRB0j6hhVPsZUx4g6RtQxqnzLbKQYUceIOkaVjzHVMaKOEXWMKt8yGylG1DGijlHlY0x1jKhjRB2jyrfMRooRdYyoY1T5GFMdI+oYUceo8i2zkWJEHSPqGFU+xlTHiDpG1DGqfMtspBhRx4g6RpWPMdUxoo4RdUwPVd5przHOWosRdYyoY1T5GFMdI+oYUcf0cNorAPVT5QHao8oDTAVVHqA9qjzAVFDlAdqjygNMBVUeoD2qPMBUUOUB2qPKA0wFVR6gPao8wFRQ5QHa47nyQ+KprnOdHawAACAASURBVDGijhF1jCofY6pjRB0j6pgenivvtNcYZ63FiDpG1DGqfIypjhF1jKhjejjtVZWPsZFiRB0j6hhVPsZUx4g6RtQxqnzLbKQYUceIOkaVjzHVMaKOEXWMKt8yGylG1DGijlHlY0x1jKhjRB2jyrfMRooRdYyoY1T5GFMdI+oYUceo8i2zkWJEHSPqGFU+xlTHiDpG1DGqfMtspBhRx4g6RpWPMdUxoo4RdYwq3zIbKUbUMaKOUeVjTHWMqGNEHaPKt8xGihF1jKhjVPkYUx0j6hhRx/RQ5Z32GuOstRhRx4g6RpWPMdUxoo4RdUwPp70CUD9VHqA9qjzAVFDlAdqjygNMBVUeoD2qPMBUUOUB2qPKA0wFVR6gPao8wFRQ5QHao8oDTAVVHqA9nis/JJ7qGiPqGFHHqPIxpjpG1DGijunhufJOe41x1lqMqGNEHaPKx5jqGFHHiDqmh9NeVfkYGylG1DGijlHlY0x1jKhjRB2jyrfMRooRdYyoY1T5GFMdI+oYUceo8i2zkWJEHSPqGFU+xlTHiDpG1DGqfMtspBhRx4g6RpWPMdUxoo4RdYwq3zIbKUbUMaKOUeVjTHWMqGNEHaPKt8xGihF1jKhjVPkYUx0j6hhRx6jyLbORYkQdI+oYVT7GVMeIOkbUMap8y2ykGFHHiDpGlY8x1TGijhF1TA9V3mmvMc5aixF1jKhjVPkYUx0j6hhRx/Rw2isA9VPlAdqjygNMBVUeoD2qPMBUUOUB2qPKA0wFVR6gPao8wFRQ5QHao8oDTAVVHqA9qjzAVFDlAdrjufJD4qmuMaKOEXWMKh9jqmNEHSPqmB6eK++01xhnrcWIOkbUMap8jKmOEXWMqGN6OO1VlY+xkWJEHSPqGFU+xlTHiDpG1DGqfMtspBhRx4g6RpWPMdUxoo4RdYwq3zIbKUbUMaKOUeVjTHWMqGNEHaPKt8xGihF1jKhjVPkYUx0j6hhRx6jyLbORYkQdI+oYVT7GVMeIOkbUMap8y2ykGFHHiDpGlY8x1TGijhF1jCrfMhspRtQxoo5R5WNMdYyoY0Qdo8q3zEaKEXWMqGNU+RhTHSPqGFHH9FDlnfYa46y1GFHHiDpGlY8x1TGijhF1TA+nvQJQP1UeoD2qPMBUUOUB2qPKA0wFVR6gPao8wFRQ5QHao8oDTAVVHqA9qjzAVFDlAdqjygNMBVUeoD2eKz8knuoaI+oYUceo8jGmOkbUMaKO6eG58k57jXHWWoyoY0Qdo8rHmOoYUceIOqaH015V+RgbKUbUMaKOUeVjTHWMqGNEHaPKt8xGihF1jKhjVPkYUx0j6hhRx6jyLbORYkQdI+oYVT7GVMeIOkbUMap8y2ykGFHHiDpGlY8x1TGijhF1jCrfMhspRtQxoo5R5WNMdYyoY0Qdo8q3zEaKEXWMqGNU+RhTHSPqGFHHqPIts5FiRB0j6hhVPsZUx4g6RtQxqnzLbKQYUceIOkaVjzHVMaKOEXVMD1Xeaa8xzlqLEXWMqGNU+RhTHSPqGFHH9HDaKwD1U+UB2qPKA0wFVR6gPao8wFRQ5QHao8oDTAVVHqA9qjzAVFDlAdqjygNMBVUeoD2qPMBUUOUB2uO58kPiqa4xoo4RdYwqH2OqY0QdI+qYHp4r77TXGGetxYg6RtQxqnyMqY4RdYyoY3o47VWVj7GRYkQdI+oYVT7GVMeIOkbUMap8y2ykGFHHiDpGlY8x1TGijhF1zEWi/uCDD87/SVV+SGykGFHHiDpGlY8x1TGijhF1zIWi3rt37w033PCrX/3qnJ9X5YfERooRdYyoY1T5GFMdI+oYUceMjPrkyZM7d+7suu7uu+/+9a9/ffb/S5UfEhspRtQxoo5R5WNMdYyoY0Qdc6GoX3vtte3bt3dd96d/+qdn/7wqPyQ2UoyoY0Qdo8rHmOoYUceIOuYiUf/4xz++7LLLuq47ePDgyk+q8kNiI8WIOkbUMap8jKmOEXWMqGMuHvXBgwe7rtu0adOPfvSjpZ/pXnzxxb3r88wzz8zOzj7zzDPrfB0u6YknnpidnX3iiSf6vpD2iTpG1BvhP/7H//i75/mt3/qtrutuvvnmvq+ufaY6RtQxoo65ZNS/8zu/03XdzMzMV77ylVJKt3fv3m597rrrrkcfffSuu+5a5+twSZ/85Cf37NnzyU9+su8LaZ+oY0RNe0x1jKhjRB2zmqg/+tGPdl132WWXvfHGG93s7Oy+9bn55pu7rrv55pvX+TpckqhjRB0j6o2wZ8+em8/zsY99rOu6nTt39n117TPVMaKOEXU9brrppqUef//995eJfFZ+r89fpog6RtQxoo753d/93aU/ifu+kPaZ6hhRx4i6Et/97ne7cz4rv/4X9bsbI+oYUceIOkaVjzHVMaKOEXUNnn/++Q15go3f3RhRx4g6RtQxqnyMqY4RdYyoe7eBz5X3uxsj6hhRx4g6RpWPMdUxoo4Rdb829rRXv7sxoo4RdYyoY1T5GFMdI+oYUfdu3759N9xww6lTp875eVV+SEQdI+oYUceo8jGmOkbUMaKuwQcffHD+T06gyi89znJ2dnb9L8XFiTpG1DGijtmzZ0/XdXv27On7QtpnqmNEHSPqmJMnTy4sLJw8eXKVv34CVR6A+nlTDaB+CwsLs7OzCwsLq/z1qjzAVFDlAeqnygMwgioPUD9VHoARVHmA+lVU5Z9++umu81eFDfTmm2/eeeedW7Zs2bJlyy233PLKK6/0fUXN2r9//7XXXrt58+Zt27bt2bPnF7/4Rd9X1L5vfOMbbiCTtfoq/9Zbb913331XXHHFpk2brr766j/6oz966623AlfYGDOc4f4coG8k1VLlT5w4ccUVV7iLbZy33357x44d3Vk2bdr08ssv931dDfrSl77U/aYrr7zyjTfe6Pu6WvbSSy/NzMy4gUzW6qv89ddff87M79q1a3FxMXCRzTDDGe7PAfpGWC1V/tZbb136/d6g1+drX/ta13W33377O++8c/r06aXbmSdGT9zs7GzXdXfcccfPf/7zUsri4uJf/MVfdF13zz339H1pzXrhhReWzqZ2A5msVVb5w4cPd1338Y9//OjRo6WUv/3bv136U/z555+PXGYLzHCG+3OGvhFWRZV/7LHHVv7qthGvTynlqquu6rruzTffXPo/T5061XXd5s2b+72q9tx0001d17333nsrP7O4uNh13datW3u8qoZ9/etfn5mZ+cxnPuMGMnGrrPJf/vKXu6578sknV35mqR498MADG3yBjTDDMe7PGfpGWP9Vfn5+fsuWLdddd527WNL777/fdd2OHTv6vpCpsPSfF/u+ijZ1XXfvvfeePn3aDWTiVlnlb7nllq7r5ufnV37m1Vdf7brulltu2eALbIQZ7pf780bTNzZaz1V+cXHx+uuv37x582uvveYuFvPee+/dc889Xdc98sgjfV9L+958882u63bv3t33hbRp5fOXbiATt8oqv/TJkDNnzqz8zFIr3b59+wZfYCPMcI/cnzeavhHQ82mvDz74YNd1Bw4cKO5iKU899dTSP6762te+1ve1TIV9+/Z1XXfw4MG+L6RxbiATt8oqv3nz5vOT99/T18AM57k/byh9o06TvMu89NJLZ/9HWHexjMcff/zGG2/csmXLzMyM3bXRXn311S1bttx66619X0j73EAmbpVVfmTyPrSwBmY4zP15o+kbdRrvLtNdQCnl/fff//jHP3755ZefOHHi7F88+UueGhdJ+3ynTp264447uq77sz/7s/B1Dt3qcz5x4sTVV1/9iU984t13381fZxtWn7YbyMR5Vz7MDCe5P8foG7WZ2F1m6bNTTz311Icv7S6WtfQZwauuuqrvC2nTiRMndu3atXPnzpW/rLKh3EAmbs2flT9z5kzns/LjM8Mx7s9h+kZVJnaXudCbbe5lSd452yBHjx7duXPntdde68+JGLeOiVtllV86FeTsJ9j87Gc/6zxGenxmOMP9uRf6Rj1U+aFaOkz37DPt5ufnu67btWtXj1fVpGeeeWbz5s033nij/26b5NYxcaus8vfff3/Xdd/73vdWfuZ73/te13Vf/vKXN/gCW2OGA9yfA/SNym3gXcZdbEMt/XF79913L52OceLEidtuu63ruv379/d9aU05duzYpk2brr322vfff7/va5kubiATt8oq/9xzzy39p/Ol016PHj26dNrr4cOHI5fZDjO80dyfM/SNyqnyQ/X222/v3LnznP/68ZnPfObsT7iyfp/97Gf9t6ZeCHniVlnlSym7du06Z9o/9alPBa6wMWZ4o7k/Z+gbYT0/V/43Xtp22mBvv/32fffdd/nll8/MzOzatWv//v2nT5/u+6Jas/Q0D39U5Al54lZf5d944409e/Zs3759Zmbm8ssvv/fee99+++3AFTbGDG809+cYfSOp59NeAajT6qs8AH1R5QEYQZUHqJ8qD8AIqjxA/VR5AEZQ5QHqp8oDMIIqD1A/VR6AEVR5gPqp8gCMoMoD1E+VB2AEVR6gfqo8ACOo8gD1q+i0VwDqocoDtGcCVf748eNzc3PHjx9f/0txceP+RY01E3WMqGNU+RhTHSPqGFFXawJVfm5ubnZ2dm5ubv0vxcWN+/Ep1kzUMaKOUeVjTHWMqGNEXS1VfkhspBhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1NVS5YfERooRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXVMD8+VV+VjbKQYUceIOkaVjzHVMaKOEXXMuFGr8kNiI8WIOkbUMap8jKmOEXWMqGNU+ZbZSDGijhF1jCofY6pjRB0j6pgeqrzTXmOctRYj6hhRx6jyMaY6RtQxoo7pocoDUD9VHqB+qjwAI6jyAPVT5QEYQZUHqJ8qD8AIqjxA/VR5AEZQ5QHqp8oDMIIqD1C/Hk57BaB+qjxAezxXfkg81TVG1DGijlHlY0x1jKhjRF0tp70OibPWYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEXS1VfkhspBhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1NVS5YfERooRdYyoY1T5GFMdI+oYUcf08Fx5VT7GRooRdYyoY1T5GFMdI+oYUceMG7UqPyQ2UoyoY0Qdo8rHmOoYUceIOkaVb5mNFCPqGFHHqPIxpjpG1DGijumhyjvtNcZZazGijhF1jCofY6pjRB0j6pgeqjwA9VPlAeqnygMwgioPUD9VHoARVHmA+qnyAIygygPUT5UHYARVHqB+qjwAI6jyAPXr4bRXAOqnygO0x3Plh8RTXWNEHSPqGFU+xlTHiDpG1NVy2uuQOGstRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEXS1VfkhspBhRx4g6RpWPMdUxoo4RdUwPz5VX5WNspBhRx4g6RpWPMdUxoo4Rdcy4UavyQ2IjxYg6RtQxqnyMqY4RdYyoY1T5ltlIMaKOEXWMKh9jqmNEHSPqmB6qvNNeY5y1FiPqGFHHqPIxpjpG1DGijumhygNQP1UeoH6qPAAjqPIA9VPlARhBlQeonyoPwAiqPED9VHkARlDlAeqnygMwgioPUL8eTnsFoH6qPEB7PFd+SDzVNUbUMaKOUeVjTHWMqGNEXS2nvQ6Js9ZiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRx/TwXHlVPsZGihF1jKhjVPkYUx0j6hhRx4wbtSo/JDZSjKhjRB2jyseY6hhRx4g6RpVvmY0UI+oYUceo8jGBqf7lL8vdd5f/+l/LK6/0/+Pv/m7jvtFLcAOJEXVMD1Xeaa8xzlqLEXWMqGNU+ZjAVD/0UOm68m/+Tem6/n988pMb941eghtIjKhjeqjywFhmZ8udd5Zt28qmTWXLlnLLLeXJJyf8Jd59t2zaVLquPPzwhF+Z4VLlm/Hzn5dNm8qOHeUrXyn33tv/j//8n/tOBBqiykPVHn549Nta99wzya/y2GN6POdS5Ztx992l68p3vtP3dQAbQJWHer30Uum6snlzeeSR8s47pZTy3nvlm99cfgf98ccn9oV279bjOZcq34ajR0vXleuu6/s6gI2hykO97rxz9HtpBw+WrivXX9/HNTE1aq7yjzxSbrutHDzY93UMwac+VbquvPBC39cBbAxVHuq1bVvpunLmzLk/v7i4/G79BC19bgdWVFvlf/CD5XGd7BZo0lJWd9zR93UAG0aVh+E5c6Z0Xdm2bZKvqcpzjmqr/AMPLI/r7/1e35dSt1OnylVXlZmZ8vrr6S/97rvlj/+4PPVU+uvCFOrhtFdgnZ58snRd+eIXJ/maqjznqLbKv/12ue++8rWvlXff7ftS6rZvX+m68sAD0S/6+utl//5yxRWT/9f5wER4rvyQeKprTDLq998v11xTtm0rP/vZJF92KFXeVMdUW+XbsxFTfeJE2bq1XH554i88zz9fPvvZcvPNy58J3KAHbU2EG0iMqKvltNchcdZaTCzqxcVy663lyivLyy9P+JWHUuVNdYwqH7MRU33PPaXryoEDE3zJC3rhhfL7v1/uuaccOFCOHi2bN9db5d1AYkRdLVV+SGykmFjUd99ddu4sJ05M/pVVec6hysdMfKrn5krXlU98oiwuTuolx6DKU0RdMVV+SGykmPVEPfIEqPOdOVM+97mydeuq/gXbKl/z/CX1M9UxtVX5xcXy9NPl/vvLgQPljTf6vpqJmvhU33hj6bry7LOTer3xqPIUUVdMlR8SGylmlXX54msvUrvfeafcemvpuvKtb03sNUcuqZ+pjlnPVE/c22+X3bvL7t3LnxvZvLk880zf1zQ5k53qp58uXVduu20iL7YWqjxF1BVT5YfERorZ0NLz8svl6qtL15VHHtmQ119ST2m7OFMdU0+VP326fPrTyw9Hf+ed5au68srVLv/yl8vtt6/rxyr/Cr1mq5nqxcXy0kuXfqnTp8s115SZmfLaa5O6urGp8hRRV0yVHxIbKWbjSs/hw2Xz5rJ9ezl8ePIvfrZKStslmeqYeqr8/v1l69byzjuljH8+1OJi2bFj9H+nWv2PW2/d0O/v0lP96qvluutK15VL/sn5yCOl68p99032AsejylNEHdTDc+VV+RgbKWaDSs/8fNmypVx5ZZmfn/Arn6+S0nZJpjqmnip/1VXlj/5o+X+/7bblq5rsuQr9ushULy6W/fvLzExZzalY77xTtm0r27Yt/7WnL6o8RdRBPZz2qsrH2EgxG1R6Pve5S7xZOEGVlLZLMtUx6xyz554r3//+Gn+c8+fDQw8t/zvXX/ziw1L7k5+s9xvcCGv7rr/97X/8+tfnDx/+3+e82uuvl+uvLzffXL7whQ9/Oy7yjd9774Z/Em81VHmKqINU+ZbZSDEbVOVX/kRU5VeY6pj1jNmpU+v6QMvOnaNfdv/+5V9wzTXr+c42yjq/66uvPnPOC37nO+Xxx0sp5a23ytaty79s9+7RX/2115aTWc0DKCf4F63zqfIUUQeNXeWXPpFzIRf6pM7Zq44dO3bkyJFjx46NtWptX2vKV508eXJ+fn5+fr7aKxzoqrH+eB7Q9zWIVfPz83Nzc/Pz89Ve4RBXvfjii2NN9Wq+1lNP/b/f/vY/rvz4y7/85che+Jd/+cuzf9nSj7/5m/898vu65prlC3jooVO1ZbhkdvZi39eF0lj6xS+99M8X+Vp/+IcnV/L/q78acYU33niq68rjj//DJb+v9f+V4+JpnF/l65n548ePj1xbzxU2s2rlXl3tFTazan5+fukt8lWu6hYWFmYvbOECfyewyqqWVo31x96Avi+rpnbV3r1711/lN/r7euGF5a8+M1OOHPnFWF/r9Oly+nT5+79/4/nn/68L/fj7v39j6Zed8+N//a83NvT7WuWqZ599afv2xaUErrvu3FX/5b+82nXl+uv/v1V+rdnZ5U/1XOjHt7/9jxf6LNB/+2//98W/r/OrfIUzb5VVzaxa+rTL6ldN4F15q6wa+irvylvV2KrZMf+C2sv3tfRE+a4rt9++vOo//aeTf/AH71181USeYHPDDRf8jwDJ36+HHjq1cknf//6Hq/7pn05ec80HXVdW/mtGX1dY/7vyVlnV3qqx35Uf+Sow5S5ecWCIqprq994rmzYtX8/TT5dSyunTZfv28o1vXHrt/fev97nyjz220d/fqpw6Va66ajmEXbs+/Ez8t75Vuq7s2dPrxZ2l5s/KQ3sW8v/sFdpTVemBiahqqg8fXr6YTZvKmTOllLJ/f9m+vbz1Vt9XlvXnf/7h78vSXzDefbdcfnnZurWcONH3xf0rVR6SVHmYgKpKD0xEVVP91ltly5alz7qUUsozz5SuK88+2/dlxS0ulp07l39fduwop06Vr361dF3Zv7/vKzuLKg9JqjxMQFWlByaitql+7rly9dWl68oVV5QdOzb8/ONqHTr04W/Nl75UZmbKxz9eTp/u85LOnPmNfxq78lGom2768Ceff77PK4SGncyf9grtqa30wPrVOdWvvlqOHl3Vo9Mbdu21v/Gvcn/wg56v5/33y+bNZevW5bNmz/+xdWu57baeLxJYMoGb+vHjx+fm5o4fP77+l+Lixv2LGmtWZ+lpkqmOMdUx40710ueLln5c6MQoRnIDiRF1tZz2OiTjfnyKNRN1jKhj9u7d23Xdvn37+r6Q9q1hqnfvXq7yr7yyYZfVIjeQGFFXS5UfEhspRtQxoo5R5WPWMNWzs6Xryhe+sGHX1Cg3kBhRV0uVHxIbKUbUMaKOUeVj1jbVx44tP5eT1XMDiRF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEHdPDc+VV+RgbKUbUMaKOUeVjTHWMqGNEHTNu1Kr8kNhIMaKOEXWMKh9jqmNEHSPqGFW+ZTZSjKhjRB2jyseY6hhRx4g6pocq77TXGGetxYg6RtQxqnyMqY4RdYyoY3qo8gDUT5UHqJ8qD8AIqjxA/VR5AEZQ5QHqp8oDMIIqD1A/VR6AEVR5gPqp8gCMoMoD1K+H014BqJ8qD9Aez5UfEk91jRF1jKhjVPkYUx0j6hhRV8tpr0PirLUYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbUMT08V16Vj7GRYkQdI+oYVT7GVMeIOkbUMeNGrcoPiY0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjeqjyTnuNcdZajKhjRB2jyseY6hhRx4g6pocqD0D9VHmA+qnyAIygygPUT5UHYARVHqB+qjwAI6jyAPVT5QEYQZUHqJ8qD8AIqjxA/Xo47RWA+qnyAO3xXPkh8VTXGFHHiDpGlY8x1TGijhF1tZz2OiTOWosRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0Qd08Nz5VX5GBspRtQxoo5R5WNMdYyoY0QdM27UqvyQ2Egxoo4RdYwqH2OqY0QdI+oYVb5lNlKMqGNEHaPKx5jqGFHHiDqmhyrvtNcYZ63FiDpG1DGqfIypjhF1jKhjeqjyANRPlQeonyoPwAiqPED9VHkARlDlAeqnygMwgioPUD9VHoARVHmA+qnyAIygygPUr4fTXgGonyoP0B7PlR8ST3WNEXWMqGNU+RhTHSPqGFFXy2mvQ+KstRhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1NVS5YfERooRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtQxPTxXXpWPsZFiRB0j6hhVPsZUx4g6RtQx40atyg+JjRQj6hhRx6jyMaY6RtQxoo5R5VtmI8WIOkbUMap8jKmOEXWMqGN6qPJOe41x1lqMqGNEHaPKx5jqGFHHiDqmhyoPQP1UeYD6qfIAjKDKA9RPlQdgBFUeoH6qPAAjqPIA9VPlARhBlQeonyoPwAiqPED9ejjtFYD6qfIA7fFc+SHxVNcYUceIOkaVjzHVMaKOEXW1nPY6JM5aixF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEXS1VfkhspBhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1NVS5YfERooRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRB3Tw3PlVfkYGylG1DGijlHlY0x1jKhjRB0zbtSq/JDYSDGijhF1jCofY6pjRB0j6hhVvmU2UoyoY0Qdo8rHmOoYUceIOqaHKu+01xhnrcWIDrDTzwAAD+xJREFUOkbUMap8jKmOEXWMqGN6qPIA1E+VB6ifKg/ACKo8QP1UeQBGUOUB6qfKAzCCKg9QP1UegBFUeYD6qfIAjKDKA9Svh9NeAaifKg/QHs+VHxJPdY0RdYyoY1T5GFMdI+oYUVfLaa9D4qy1GFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEXS1VfkhspBhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1DE9PFdelY+xkWJEHSPqGFU+xlTHiDpG1DHjRq3KD4mNFCPqGFHHqPIxpjpG1DGijlHlW2YjxYg6RtQxqnyMqY4RdYyoY3qo8k57jXHWWoyoY0Qdo8rHmOoYUceIOqaHKg9A/VR5gPqp8gCMoMoD1E+VB2AEVR6gfqo8ACOo8gD1U+UBGEGVB6ifKg/ACKo8QP16OO0VgPqp8gDt8Vz5IfFU1xhRx4g6RpWPMdUxoo4RdbWc9jokzlqLEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1eqW/h3DhVzo3zecverIkSOzs7NHjhwZa9XavtaUr1pYWFj6W1O1V9jMqoXz7lm1XWEzq1beC6j2Coe46sUXX9x7lgMHDjzxxBN79+7ds2fPoUOHarjCtlct/OsNpNorbGbV/Pz8yH5ZzxU2s2rlXl3tFU7tqm5hYWH2whYu8Ncvq3pZdfElNVxhM6sWzqvytV2hVVZdZNXSG/Ar9uzZU9sVtr1q5QZS7RU2s2qpX56/tp4rtMqqcVct/QV19au8Kz+kVQvelU+tWvCufGqVd+U3YtXs7Oy+szz66KOHDh3at2+fd+Uzqxa8K59a5V352CrvysdWLU31hcreiHflR77KWHxWPmbBJ9VSRB0j6hiflY8x1TGijhF1zLhRq/JDYiPFiDpG1DGqfIypjhF1jKhjeqjyTnuNOemstRRRx4g6RpWPMdUxoo4RdUwPVR6A+qnyAPVT5QEYQZUHqJ8qD8AIqjxA/VR5AEZQ5QHqp8oDMIIqD1A/VR6AEVR5gPqN+7AgVR5gKqjyAO3xXPkh8VTXGFHHiDpGlY8x1TGijhF1tZz2OiTOWosRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0Qd08Nz5VX5GBspRtQxoo5R5WNMdYyoY0QdM27UqvyQ2Egxoo4RdYwqH2OqY0QdI+oYVb5lNlKMqGNEHaPKx5jqGFHHiDqmhyrvtNcYZ63FiDpG1DGqfIypjhF1jKhjeqjyANRPlQeonyoPwAiqPED9VHkARlDlAeqnygMwgioPUD9VHoARVHmA+qnyAIygygPUr4fTXgGonyoP0B7PlR8ST3WNEXWMqGNU+RhTHSPqGFFXy2mvQ+KstRhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1NVS5YfERooRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtQxPTxXXpWPsZFiRB0j6hhVPsZUx4g6RtQx40atyg+JjRQj6hhRx6jyMaY6RtQxoo5R5VtmI8WIOkbUMap8jKmOEXWMqGN6qPJOe41x1lqMqGNEHaPKx5jqGFHHiDqmhyoPQP1UeYD6qfIAjKDKA9RPlQdgBFUeoH6qPAAjqPIA9VPlARhBlQeonyoPwAiqPED9ejjtFYD6qfIA7fFc+SHxVNcYUceIOkaVjzHVMaKOEXW1nPY6JM5aixF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEXS1VfkhspBhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1NVS5YfERooRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRB3Tw3PlVfkYGylG1DGijlHlY0x1jKhjRB0zbtSq/JDYSDGijhF1jCofY6pjRB0j6hhVvmU2UoyoY0Qdo8rHmOoYUceIOqaHKu+01xhnrcWIOkbUMap8jKmOEXWMqGN6qPIA1E+VB6ifKg/ACKo8QP1UeQBGUOUB6qfKAzCCKg9QP1UegBFUeYD6qfIAjKDKA9Svh9NeAaifKg/QHs+VHxJPdY0RdYyoY1T5GFMdI+oYUVfLaa9D4qy1GFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEXS1VfkhspBhRx4g6RpWPMdUxoo4RdbVU+SGxkWJEHSPqGFU+xlTHiDpG1DE9PFdelY+xkWJEHSPqGFU+xlTHiDpG1DHjRq3KD4mNFCPqGFHHqPIxpjpG1DGijlHlW2YjxYg6RtQxqnyMqY4RdYyoY3qo8k57jXHWWoyoY0Qdo8rHmOoYUceIOqaHKg9A/VR5gPqp8gCMoMoD1E+VB2AEVR6gfqo8ACOo8gD1U+UBGEGVB6ifKg/ACKo8QP16OO0VgPqp8gDt8Vz5IfFU1xhRx4g6RpWPMdUxoo4RdbWc9jokzlqLEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRV0uVHxIbKUbUMaKOUeVjTHWMqGNEHdPDc+VV+RgbKUbUMaKOUeVjTHWMqGNEHTNu1Kr8kNhIMaKOEXWMKh9jqmNEHSPqGFW+ZTZSjKhjRB2jyseY6hhRx4g6pocq77TXGGetxYg6RtQxqnyMqY4RdYyoY3qo8gDUT5UHqJ8qD8AIqjxA/VR5AEZQ5QHqp8oDMIIqD1A/VR6AEVR5gPqp8gCMoMoD1K+H014BqJ8qD9Aez5UfEk91jRF1jKhjVPkYUx0j6hhRV8tpr0PirLUYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbUMT08V16Vj7GRYkQdI+oYVT7GVMeIOkbUMeNGrcoPiY0UI+oYUceo8jGmOkbUMaKOUeVbZiPFiDpG1DGqfIypjhF1jKhjeqjyTnuNcdZajKhjRB2jyseY6hhRx4g6pocqD0D9VHmA+qnyAIygygPUT5UHYARVHqB+qjwAI6jyAPVT5QEYQZUHqJ8qD8AIqjxA/Xo47RWA+qnyAO3plrr/hVzo7wRnrzp27NiRI0eOHTs21qq1fa0pX3Xy5Mn5+fn5+flqr7CZVef/nbi2K2xm1fz8/Nzc3Pz8fLVXOMRVL7744t6zHDhw4Iknnti7d++ePXsOHTpUwxW2vWrlBlLtFTaz6vjx4yPX1nOFzaxauVdXe4VTu6pbWFiYvbCFC3xSx6peVl18SQ1X2MyqhfM+qVbbFVpl1UVWLb0Bv2LPnj21XWHbq1ZuINVeYTOrls6bP39tPVdolVUbvWoC78ofOXJkdnb2yJEjY61a29ea8lULCwuzs7Nzc3PVXmEzqxbOq/K1XWEzq5b+JJ6bm6v2Coe4anZ2dt9ZHn300UOHDu3bt8+78plVC/96A6n2CptZNT8/P7Lc1HOFzaxauVdXe4VTu2oCn5Vf+d1d/0txcQtj/qNm1kzUMaKO8Vn5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtQxJ/PPlVflY2ykGFHHiDpGlY8x1TGijhF1zLhRq/JDYiPFiDpG1DGqfIypjhF1jKhjVPmW2Ugxoo4RdYwqH2OqY0QdI+qYHqr88ePH5+bmjh8/vv6X4uLG/fgUaybqGFHHqPIxpjpG1DGijumhygNQP1UeoH6qPAAjqPIA9VPlARhBlQeonyoPwAiqPED9VHkARlDlAeqnygMwgioPUL8eTnsFoH6qPEB7PFd+SDzVNUbUMaKOUeVjTHWMqGNEXS2nvQ6Js9ZiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1tVT5IbGRYkQdI+oYVT7GVMeIOkbU1VLlh8RGihF1jKhjVPkYUx0j6hhRx/TwXHlVPsZGihF1jKhjVPkYUx0j6hhRx4wbtSo/JDZSjKhjRB2jyseY6hhRx4g6RpVvmY0UI+oYUceo8jGmOkbUMaKO6aHKO+01xllrMaKOEXWMKh9jqmNEHSPqmB6qPAD1U+UB6qfKAzCCKg9QP1UegBFUeYD6qfIAjKDKA9RPlQdgBFUeoH6qPAAjqPIA9evhtFcA6qfKA7THc+WHxFNdY0QdI+oYVT7GVMeIOkbU1XLa65A4ay1G1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFFXS5UfEhspRtQxoo5R5WNMdYyoY0RdLVV+SGykGFHHiDpGlY8x1TGijhF1TA/PlVflY2ykGFHHiDpGlY8x1TGijhF1zLhRq/JDYiPFiDpG1DGqfIypjhF1jKhjVPmW2Ugxoo4RdYwqH2OqY0QdI+qYHqq8015jnLUWI+oYUceo8jGmOkbUMaKO6aHKA1A/VR6gfqo8ACOo8gD1U+UBGEGVB6ifKg/ACKo8QP1UeQBGUOUB6qfKAzCCKg9Qvx5OewWgfqo8QHs8V35IPNU1RtQxoo5R5WNMdYyoY0RdLae9Domz1mJEHSPqGFU+xlTHiDpG1NVS5YfERooRdYyoY1T5GFMdI+oYUVdLlR8SGylG1DGijlHlY0x1jKhjRF0tVX5IbKQYUceIOkaVjzHVMaKOEXW1VPkhsZFiRB0j6hhVPsZUx4g6RtTVUuWHxEaKEXWMqGNU+RhTHSPqGFHH9PBceVU+xkaKEXWMqGNU+RhTHSPqGFH//6MbIDWoASI5XhFL7ZoyAAAAAElFTkSuQmCC[/img][/center][b][u]Complex Number Norm (Magnitude)[br][/u][br][/b][/b][/b][u]The norm of [/u][math]z[/math], denoted [math]|z|[/math]or [math]||z||[/math], is defined by [math]|z|=|a+bi|=\sqrt{a^2+b^2}[/math].[center][br][b][b][/b][/b][img width=242,height=196]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAmAAAAHsCAIAAAD6k4a5AAAgAElEQVR4nO3df2hU94L//zdM8omusSYwV4IEN1sqVURs6UXElTa4gdqPiooRhhKyWQmyXlu6xVt6Kb2Nmy2tELppCRLEbCNNaaTZS7Y3SK5YarDSSisbJLheyUrE+DGu8evQ73zakGbT8/njuMcxPyfznnm9J2eeDy4XOsmZvPPK+H7NmXPO+xgPWAzi8fjQ0FA8Hnc9kPAjahmilkkvapOl0eSPmzdvuh5CXhgaGurr6xsaGnI9kPAjahmilkkvagrSym9+85uCgoLDhw+7Hkj4MZXIELUMUctQkGq9vb2RSMQYE4lE/vSnP7keTsgxlcgQtQxRy1CQUidPniwoKDDGPPHEE8aYgoKCtrY214MKM6YSGaKWIWoZClLkl19+efPNN40xxpitW7cG/2+MefPNN3/55RfXAwwnphIZopYhahkKUuGnn36qrq42xhQWFra1tTU0NBhjjh492tbWVlhYaIyprq7+6aefXA8zhJhKZIhahqhlKMis+/nnnzdv3myMKSkp+fLLLz3PCwrS87wvv/xyxYoVxpjNmzf//PPPrgcbNkwlMkQtQ9QyFKRCQ0PDX/3VX/3Hf/xH8J9BQXqed/Xq1YqKiuA/kUFMJTJELUPUMhSkSPKlplMKcspXkUFMJTJELUPUMhSkA9MLElnCmiMyRC1D1DKspOMABQkAYUVBWqEgASCsKEgrFCQAhBUFaYWCBICwoiCtUJAAEFYUpBUKEgDCioK0QkECQFhRkFYoSBmuGJMhahmiluE6SAcoSBnWHJEhahmilmElHQcoSBmmEhmiliFqGQrSAQpShqlEhqhliFqGgnSAgpRhKpEhahmilqEgHaAgZZhKZIhahqhlKEgHKEgZphIZopYhahkK0gEKUoapRIaoZYhahoJ0gIKUYSqRIWoZopahIB2gIGWYSmSIWoaoZShIByhIGdYckSFqGaKWYSUdByhIAAgrCtIKBQkAYUVBWqEgASCsKEgrFCQAhBUFaYWCBICwoiCtUJAAEFYUpBUKEgDCioK0QkHKcMWYDFHLELUM10E6QEHKsOaIDFHLELUMK+k4QEHKMJXIELUMUctQkA5QkDJMJTJELUPUMhSkAxSkDFOJDFHLELUMBekABSnDVCJD1DJELUNBOkBByjCVyBC1DFHLUJAOUJAyTCUyRC1D1DIUpAMUpAxTiQxRyxC1DAXpAAUpw1QiQ9QyRC1DQTpAQcqw5ogMUcsQtQwr6ThAQQJAWFGQVihIAAgrCtIKBQkAYWWGh4f7Zzc8PDzjZnm41fnz50+ePNnwuBdeeMEY09bWlgsjZCu2Yiu2YqsMbmX6+/v7Ztff3z/jD8jPrT788EMzk3/7t3/LkRGyFVuxFVuxVaa2Yg8y1a36+vra2tqOPq6yspI9SLZiK7Ziq1BuxTFIKxyDBICwoiCtUJAyXDEmQ9QyRC3DdZAOUJAyrDkiQ9QyRC3DSjoOUJAyTCUyRC1D1DIUpAMUpAxTiQxRyxC1DAXpAAUpw1QiQ9QyRC1DQTpAQcowlcgQtQxRy1CQDlCQMkwlMkQtQ9QyFKQDFKQMU4kMUcsQtQwF6QAFKcNUIkPUMkQtQ0E6QEHKMJXIELUMUctQkA5QkDKsOSJD1DJELcNKOg5QkAAQVhSkFQoSAMKKgrRCQQJAWFGQVihIAAgrCtIKBQkAYUVBWqEgASCsKEgrFCQAhBUFaYWClOGKMRmiliFqGa6DdICClGHNERmiliFqGVbScYCClGEqkSFqGaKWoSAdoCBlmEpkiFqGqGUoSAcoSBmmEhmiliFqGQrSAQpShqlEhqhliFqGgnSAgpRhKpEhahmilqEgHaAgZZhKZIhahqhlKEgHKEgZphIZopYhahkK0gEKUoapRIaoZYhahoJ0gIKUYc0RGaKWIWoZVtJxgIIEgLCiIK1QkAAQVhSkFQoSAMKKgrRCQQJAWFGQVihIAAgrCtIKBQkAYUVBWqEgASCsKEgrFKQMV4zJELUMUctwHaQDFKQMa47IELUMUcuwko4DFKQMU4kMUcsQtQwF6QAFKcNUIkPUMkQtQ0E6QEHKMJXIELUMUctQkA5QkDJMJTJELUPUMhSkAxSkDFOJDFHLELUMBekABSnDVCJD1DJELUNBOkBByjCVyBC1DFHLUJAOUJAyTCUyRC1D1DIUpAMUpAxrjsgQtQxRy7CSjgMUJACEFQVphYIEgLCiIK1QkAAQVhSkFQoSAMKKgrRCQQJAWFGQVihIAAgrCtIKBQkAYUVBWqEgZbhiTIaoZYhahusgHaAgZVhzRIaoZYhahpV0HKAgZZhKZIhahqhlKEgHKEgZphIZopYhahkK0gEKUoapRIaoZYhahoJ0gIKUYSqRIWoZopahIB2gIGWYSmSIWoaoZShIByhIGaYSGaKWIWoZCtIBClKGqUSGqGWIWoaCdICClGEqkSFqGaKWoSAdoCBlWHNEhqhliFqGlXQcoCABIKwoSCsUJACEFQVphYIEgLCiIK1QkAAQVhSkFQoSAMKKgrRCQQJAWFGQVihIAAgrCtIKBSnDFWMyRC1D1DJcB+kABSnDmiMyRC1D1DKspOMABSnDVCJD1DJELUNBOkBByjCVyBC1DFHLUJAOUJAyTCUyRC1D1DIUpAMUpAxTiQxRyxC1DAXpAAUpw1QiQ9QyRC1DQTpAQcowlcgQtQxRy1CQDlCQMkwlMkQtQ9QyFKQDFKQMU4kMUcsQtQwF6QAFKcOaIzJELUPUMqyk4wAFCaTixx9/fOmll44dO3ZZ4r/+679c/8YIAwrSCgUJpOLtt982Qv/8z//s+jdGGFCQVihImR9//HHPnj2yXRCHfvnlF9dhZ9iNGzcKCwufeOKJv/u7vzso8eWXX7r+pREGFKQVClJGvAvi0Pj4uOuwM2z//v3GmBMnTrgeCLAwFKQVClLD3wWJRqOyXRCH/vu//9t13pl06dIlY8yGDRtcDwRYsEVWkJ9++um1a9dcj+KRHCnIXIsl49gFWbyeffZZY8xXX33leiDAgi2mghwbGzPGxGIx1wN5RFmQP/zwQ09Pz71796Y8noOxZBa7IIvXp59+aozZtWuX64EA6VhMBZlIJIwxe/fudT2QR5QF6b8TLy0tHRwcTH48B2PJLP8X/+Mf/8gVYxqZujhvbGxs1apVkUhkyisWAa6DtNfc7L31lvfgwTzfFv7rIFNpgitXrjQ1NdXV1b388ssHDhxobm7+7rvvsjckm4I8e/bsW2+9VVtbW1NTc/DgwU8//fT27duzffO1a9eCkzg+++yz5C+FuyCDXRDWHJHJVNRHjx41xrzxxhuZGFQ4ZfZVffeu19vrnTnjnT3rjY1l5CnVrl/3zpzxenq8ixe9ycn5v//CBc8YzxjvyJF5vjP8K+nM3QT379/ft29fJBLZtm3boUOHDh06tHHjRr9RnnvuuatXr1r+9LNnz65YseL69evJD6ZXkD09PRUVFf7Hhjt27CgvLw/KLxaL3b17d/omH330kf8N69atSyQSyV8KcUEm74LM+Pq+devWqVOn6uvra2tr6+rqjh8/fuvWLUeDXbBz587FYjH7V2bGZWTWvnPnzrJly0pLSx/M997+wYMHr7/++unTp1N5WvE74GzLVEGeOOGtW+c9+6y3f7+3f7+3bp1njPf8815PTyZGKdHS4lVUeEVFXmmpV1TkGeMVFnr793tXrsy1VV3dw4L8+ON5nj+vC3JycvK5554rLCzsefwVsWXLFr9XSktLp3TbQr366qvGmC+++CL5wTQK8tSpU8aYHTt2JE/lPT09paWl/lBXrlw5fdLctWuXMebJJ5+8c+fOlC+FuCCTd0GmvL7v379fW1vrJ7Z79+7Dhw/X1NREIpFIJPLqq6+OZeL984xviTJlcHBwxYoVxpju7u5sPL+NjMza/l+nubl5ju8ZHBxsbGyMRqPGmNra2rmfMNvvgJ2wjzqR8KqqvFjMm/LO8NSph81x8KDlGLNubMyrqvI2bfIuXHj04Nmz3tNPP/wV3ntv5g0TCa+w0DPG279//p+S1wXpt44xpqioKPkd5WeffRbsnNXV1dn89MOHD0+fzhZakGNjY8uWLTPGRCKR48ePJ3/piy++CIa6bt265C9NTEwUFhZWVFRMb0cvvAU5ZRck+fV97969J5980s8q+fTICxcuRCIRY8yLL75oP4AZ3xJlRCKRePrpp/3xh7Ig+/v7/Zfx5LSPyc6dO7d79+7Kysrly5cnX/05d0EK3gE7YR/1/v3eW2/N/KX6+ocFk9kPuTN+me7u3d6OHd7ExNTHHzzw1q+fqyM/+sgzxtuwIaXPk/O6IOvr64N/aa+//nrweE9PT/D4Sy+9ZPPTM1KQvb29wXiWLl065avr1q0Lvtrb2xs8fu7cudWrV8/Yjl54C3LKLkjy6/ull17yU9q6deuUrfwLQowxbW1tlgOY8S+eETt27Aj+0KEsSL+3emb6gO+rr76KxWK1tbXNzc2XLl0qKipKpSAF74CdsIx6cNAzxisq8l580bt/f+pXe3oetkskMsNX07Z6tdfUlLFn+/xzr6zMe/yo0SNXrz76FaZfyLZhg1da6t28mdIPyuuCbG1tDf6dnDp1Knj89OnTweP79u2z+ekZKchvvvkmGM+yZcumfPXll18OvvrRRx8Fj4+Njc14YNIXyoKcvgsSvL6vXLkyx27H8ePH/S+tWbPGcgxZKsijR4/6HyqGtSA///zz1N+PpliQgnfATlhH/bA/jPGmvyG8cePRVzP4OUhRkXf4cMaerbLS27/fm+M4cmXlw1/h0KHHHr90yTPGS9qPmEdeF6TneUePHt21a9e7776b/GDyv6s//OEPNj89IwXped6BAwf8N8LJRe6rqakJRhs85+Tk5Jtvvnn48OHDhw8PDAxMf8JQFuT0XZDg9X3ixIkgpcPT/qUmvyW6ePGizRiyUZA9PT2RSKSvr2/RFeTk5GQqeY6Pjz/55JORSCTFg4IpFqTgHbATlgX51VePKvDTT6d+9cqVXC/IyUkvEnk4wsdn7kfefvvhN5SXP/Z4fb33zjsL+Fn5XpDT3b59O/jnZ//xS6YK0vO88fHxiemfuP/PBX++4NDX5ORk8OAnn3wyfavwFeSMuyDB6/vQoUNzFGR3d3fw1WPHjtkMI+MFOTg4uHz58hMnToyPjy+ughwYGNiwYYMxpr+/f+5tjx07Zow5NOXd/uxSLEgv+++AnbD/NPutt7znn/defXWGY3hnzjwqyAwen81gQY6PPxphRcXM3/PJJ4++J9lsR15nQ0E+MjExcfHiRf8kt6KiovdmOwtqITJYkDO6fv168E/96aefTj67IXj87Nmz0zdMoyDr6+t32Pnggw8W+gta7oIEr+833nhjjoL8wx/+EHx13jl3bpktyB9++GHNmjUHDhzwPM+mIAV/u+SpZHJysrGx0T/1yRhTVVU1x4b3799fvnz58uXL76d8yCv1gpwus++Ancjq1b3vvvuwV559NpNPm9mPWMvKHg5ytg/IkwsylSsjZ0NBep7nDQwMBP+YjTGlpaVtbW3TT6VLQ7YLMjgAWVpaOqUbgllgxvs8LLQgJycny8rKjJ2FniZqvwsSLISR/Kna9IJsa2sLvrp79+4FDXKKzBbkjh07Nm3a5H9ykHZBav52QdSDg4MbN26srKxMPjp+Iflk/McdPHjQLHDHPb2CzMY7YCeyupKOfzWkMV5fXyafNrMFefmy9/LL3quverOdZRF8xLpy5WOP19R4O3Z4NTWp/iBW0nlofHx8fHz81q1bH330kX8qeVlZmf0611ktyOCDwbKysukV4v8WK1asmHHbHP+INeO7IA8ePCgsLJxtVo3FYsFUvmPHDpuRZ7Agjx49unLlyuBMqxz/iDVw4sSJ1tZWz/Pu3r3rX55kjNm0adOM33z16lVjzJNPPrmg96MLLcjsvQMOmY8/ftgrra0ZfubMFuS8gpN0Dhx47PHS0hlaM+NCWJDJkvc25l3y6quvvvpkdlVVVcaY1157LfnBvXv3+kPy//PK3Ks+zOLq1at+BW7ZsmXGs1X9NQRm2x/K5YLM0i5IY2Oj/1TPPv7h0eDgYNCduVOQ/ok5yR8vL5aCTJZ8P84Zrw3dtm2bWfiBwDT2ILP0Djg0Jie9lhavsNBbtco7cybzz68syFu3Hn2+OmVyjUY9Y7xsL4Mf8oKcnJxMnjG/+eabOb7TWFu/fv1Cf6kbN274H5odOXJktjfC/oeTB2dZEiOXCzJ7uyB+lRpjGhsb/e8fHBxct27da6+9Fvw55v2INY23RFPM+5bo+vXr/ok5yQ8uxoJ88OBBsNjT9Dur+JdbPP/88wt9WptjkN4C3wGH26lT3r59XmWlt3SpZ4y3d2+2lmNVFuTrrz9sx+k/cdMmz5gFfMSanpAU5NWrV48dO3b69Onpc+tzzz0X/BOa+0h+X1+f5R7kjJdhzOHmzZurVq2KRqO9j1/OU1tbe+7cueA//eviP51+HrfnebldkMkyvgty5syZbdu2RSKRoqKi0tLSVatWff7558lnsc67OIvNmyHf3G+Jkk/MSbYYC9LzvHfffTcY9idJJ1RPTk76CwOl8QmKZUGm/g449MbGvAsXvAsXvLNnvQ8+8Fat8qJRr7Ex8zUpK8hr1x5eBLJlywzn6L700gwXR2ZcGAryypUrwT+zmmnvKPxi89l85pbxY5C3bt1atWrVhg0bbj6+FMT9+/en/CB/IdYLFy7cvXt3+sG59Apy3MKM16jMK0u7IJOTkzdu3AgyTN6lmPdjtzTeEk0x91uiHTt2bNy48ccff5wSoP8n833++ef+g6kfSNP/7Xz+2vH+sNesWRMM+IMPPjDpnkeaYkFm5B1wXrl3z3vySc8Yb/16b5Y1uNKkKcjJyYf7iBs3znwrqx07PGO8aReTZ1gYCvLIkSPBv5BIJDLlq8FqjcaY/aksajuLzBbk7du3y8vL9+7dm5i2yJK/qlbyIat9+/YZY27fvn369OmVK1da3g8yI2dCVlZWLvRX9jK6C3L58uUvvvhi+nTpT9Y+y8U57Y9BLijSMykcL3L4t/MFd5UxxrS0tHj/875n2bJlsy2FOLdUClLzDjh8zp59+PnkmjWzruU2xZ073pkz8/yvsNDbsWOe75n9NINUvfaaZ4y3adOsi+T5BZnta1/DUJDvvfde8C9k5bSzmoJ/WiaFXYo5ZLAg79y5s3r16sLCwunXqFVWVvoDHk+6osM/w6W2ttbflYzFYsnPlsYe5OHDhy2vpfMnx4XK1C7IJ5984j/J9HNig7NYpy/TulD2Bdk9i+QDpW+++ab/4ByrCU4ZlZO/nW9yctK/U5sxpqysbGxszP9dGhsb03vCVApS8w44lIKrDN98M6Xvf+65RyfFWP7P5tBBW9vD23X98MOs37Nr18KWmktPGAry+vXr/pnfK1eunLI+cvKaXhs3brQ5HTxTBXnnzp3gThSzWb58efIm77zzTvJXp6wyuliOQfrS3gVJvowp+VqO5L9pIpEITgW6dOmS5VCzt1h5UPBZen5Lc18xFqwbbow5cOBAJBJZvXr1eLq3eEilIDXvgJ3I6nWQnuft3fuwrpYuneEw3nR377rfg/TXWN+3b57bhvi/Wuq3f83r6yBPnz5dWFi4fv36jz/+OLhB67lz54L9lXXr1qX3EVAgIwV5//794D5Hc5iyY3T58uXkr065On5xFWTauyDJC2EEuxRTggruEJn2Dk2yvC3IedccWb9+ffILcrbTx1KRSkFq3gE7YbmSTiLhffGFN8faGwcOPNqly9QlH1k9BvnNN97SpV7ScvSz2r9/YWvr5PtKOlevXq2trfVPaSssLAzObVu9enVjY6P9HXQzUpBnzpyZtx1nnCyCBUg3bdp079695C8troL00t0FSX5937hxY/ny5StXrgx2E69du+afUVxUVJR8IxQbGS/I4KqS4BoVk3QS0O3btzP1gyzNO5Uknyo820U7s5mYmEg+yyn4d7p169bgweRTuH2Cd8BO2BTk/fuPPkGdbc85uCWk5WeeybJXkNeueaWls65aXl//2H/GYt60GwbOJd8L0jc5Odnf39/T0/PFF1/09vameFwnFdleam5e169fn/Gsk0VXkF5auyBTXt/fffedv9JY8GaosLDwwIEDGbxrbsYLcvfu3cuWLVs+k6VLl6Zyko5GKlPJpk2b/L/d5cuXF/TkiUSiqKhothyWL1++bNmyGW9cle13wE7MuAx3ik6ffrT5bAfcF9Ee5J07Xnn5rHeaHBvzVq167JFYbOr9PeZGQWaX84KczWIsyDR2QWZ8fQ8ODvb09PT09Fy8eDHjH69l7yPWHJfKVOJ/tvnyyy+rBvVQ9t4BO2FTkMk3g5xtpe/gGGQkMtcJLwuSjYJ88MBbv947fnzWb+jp8abMcC+//Oi3vnx5/tN0Kcjs8g+VTbnCnYJM20J3QbJ634MZUZBzf9uVK1dsrqqEZ1eQ/jJsS5d6VVXe41d+PbJy5cPnz+CKMxkvyPFx7/nn51ozdnLSq6+fuiZAfb23apX34IE3MeGtX++99to8P4WCzC7/gMeUaxApyLQtdBdEX5AzviXKB/qo85ZNQXqeV1/vVVbOenpqd/fDJ49GM7lWQMYL0t/NXb585v8tXfpwPZ0px2FaWx++P1ixwjPGKy2d56dQkA5QkDYWtAuin7VnfEuUDyhIGcuCTCS8LVu8ysoZrna4c8dbtephc1hf8fSYzBbkwYOpXlg55cyta9ce++qWLfP8IArSAQpShllbhqhlLAvS87yxMe/IEa+w0Nu71ztxwuvt9Xp7vaamhye4btvmPb6QZQZksCAnJhaw8sD089yDU5CKiryZ7iX/mDQL0r98cjazXVbpZKs///nPxphdu3Y5GeH58+ebm5sbHvfCCy8YYz788EN9GoHkWHLq75XZrYZmen3n1AhDs1V/f39fX19/f3/OjnCRbrWglWgW9LO+/fbWW289+N//+/9WVf28ZYu3a5d35Ij33XdZ+b3+1//6pbb2/3eV4RR//GPi+PF7Fy8Oz7vVUNIEkvrPMv5msxmapW9dbdXY2PjJJ5+4GmFdXd2Mly2eOnXKSRrTY8m1v1cGt5pxIYycGiFbsdXcWy2oIHP29yosnNy79/+4yjDtrZInkNS3Wkx7kG636uvr+/DDD48+rrKy0vkeJFuxFVstiq2ytwep/L1WrZr4/e//P1cZirdaTMcgc1AuHIMEsLgMDAwUFxcvWbLE/hikXrrL7i5Ki+ovk3soSABp6OrqGhgYWIwFmVf4y1ihIAGkjYLMcfxlrFCQANJGQeY4/jJWKEgAc+js7JztVBGPgsx5/GWsUJAAZvMP//APxphYLDbbN1CQOY6/jBUKUmbG6yCRDUSdKf39/eXl5S0tLbN9A1HLpBc1BWmFgpQZYv0zFaLOoLlXGyZqmfSipiCtUJAyTCUyRC1D1DIUpAMUpAxTiQxRyxC1DAXpAAUpw1QiQ9QL1dXVtXXr1sS8d7WfhqhlKEgHKEgZphIZok7d6Ojonj17/JsWtLa2LnRzopahIB2gIGWYSmSIOkUdHR3RaNQYU1xc3NzcnPrdvwNELUNBOkBByjCVyBD1vEZGRmKxmL/juHXr1sHBwfSeh6hlKEgHKEgZphIZop5bZ2env+O4ZMmS1tbWNHYcA0QtQ0E6QEHKMJXIEPVshoeHt2/f7u84VlVVpb3jGCBqGQrSAQpShjVHZIh6Rq2trcXFxTZHHKcjahlW0nGAggRCb2hoqKqqyv6IIxYdCtIKBQmEW2tra0lJiX/EMVM7jlgsKEgrFCQQVsPDw8k7jhwpzEMUpBUKEgir4eHhaDRaXFxseaoqFi8K0goFCYRYd3f3tWvXXI8CzlCQVihIAAgrCtIKBQkAYUVBWqEgZbhiTCYPo+7u7nby++Zh1K5wHaQDFKQMa47I5FvUzc3NBQUF1dXV+h+db1E7xEo6DlCQMkwlMvkW9cjISEVFRUtLi/5H51vUDlGQDlCQMkwlMnkYtaurOPIwalcoSAcoSBmmEhmiliFqGQrSAQpShqlEJsRR59r1/iGOOtdQkA5QkDJMJTJhjbq7u/uZZ54ZHR11PZBHwhp1DqIgHaAgZZhKZMIXdTwer66u9ldVdXIyzmzCF3XOoiAdoCBlmEpkQhZ1Z2dneXl5Zu/jmCkhizqXUZAOUJAyTCUyoYl6ZGQkFovl8n0cQxN17qMgHaAgZVhzRCYcUXd3d5eVlfn3cWxqasqpHcdAOKJeFFhJxwEKEsg18Xg82HHcvHlzDu44YrGgIK1QkEBO6e7ujkajxpiCgoL3338/N3ccsVhQkFYoSCBHjI6OxmKxgoICY8yvf/1r7uMIexSkFQoSyAWdnZ3+EUd/x3FsbMz1iBAGFKQVChJwKx6P19TUBEccBwYGXI8I4UFBWqEgAbcSicRTTz21ZMmShoaGRCLhejgIFQrSCgUJOHf+/Pn+/n7Xo0AIUZBWKEgZrhiTIWoZopbhOkgHKEgZ1hyRIWoZopZhJR0HKEgZphKZnI26u7s7ZDtbORt1+FCQDlCQMkwlMrkZdXNzc0FBQXV1teuBZFJuRh1KFKQDFKQMU4lMbkY9MjJSUVGRUzerspebUYcSBekABSnDVCKTs1GHb924nI06fChIByhIGaYSGaKWIWoZCtIBClKGqUTGedTh21OcjfOo8wcF6QAFKcNUIuM26u7u7rVr1+bJPap4VctQkA5QkDJMJTKuoo7H49XV1f6qqm+//bb4pzvBq1qGgnSAgpRhzREZJ1F3dnaWl5cbY5YsWdLU1JQnn7LyqpZhJR0HKEjA0sjISCwW83cct27dyu04kDsoSCsUJGCju7vbv4/jkiVLTp48mSc7jlgsKEgrFCSQnng8Huw4VlVVDQ8Pux4RMBUFaYWCBNLQ3d0djUaNMcXFxc3Nzew4IjdRkCvVpJAAABfMSURBVFYoSGBBRkdHY7FYQUGBMWb79u15cjkHFikK0goFCaSus7PTP+JYXFzc2trKjiNyHAVphYIEUhGPx2tqavwjjjt37uTKPywKFKQVClKGK8ZkshF1PB4vKysrKSnp6OhgxzHAq1qG6yAdoCBlWHNEJktRf//99yMjI5l9zsWOV7UMK+k4QEHKMJXIELUMUctQkA5QkDJMJTJELUPUMhSkAxSkDFOJjGXUXV1dXLyRIl7VMhSkAxSkDFOJjE3U//RP/2SM+fWvf82ZOKngVS1DQTpAQcowlcjYRJ1IJJ566qn333+fgkwFr2oZCtIBClKGqUTGMmqqMXW8qmUoSAcoSBmmEhmiliFqGQrSAQpShqlEJsWo2VO0x6tahoJ0gIKUYc0RmVSi7u7uXrt2LTO7JV7VMqyk4wAFiXwTj8erq6v9VVXff/9918MBsoiCtEJBIq90dnaWl5cbY6LRaFdXl+vhANlFQVqhIJEnRkZGYrGYv+O4Z8+e4eFh1yMCso6CtEJBIh90d3f793EsKyvjdhzIHxSkFQoS4RaPx4Mdx507d3I7DuQVCtIKBYkQ6+7ujkaj/o5je3u76+EAahSkFQoSoTQ6OhqLxQoKCvwjjqOjo65HBDhAQVqhIGW4YkzmX/7lX371q18ZY4qLiznimFW8qmW4DtIBClKGNUcE4vF4TU2Nf8Rx165dnKqabbyqZVhJxwEKUoapRMC/F0dpaenvfvc7ohbgVS1DQTpAQcowlWj09/d/++23RK3Bq1qGgnSAgpRhKpEhahmilqEgHaAgZZhKZIhahqhlKEgHKEgZppLM6urq+v7772f8ElHLELUMBekABSnDVJJBHR0dBQUFa9eunfESDqKWIWoZCtIBClKGqSSDJiYmnnnmmbfffpuCdIuoZShIByhIGaaSzJrj8n+iliFqGQrSAQpShjVHZIhahqhlWEnHAQoSOY6F4oC0UZBWKEjkst7e3rVr1w4ODroeCLAoUZBWKEjkJv92HP6qqm+//bbr4QCLEgVphYJEDurq6iorKzPGlJSUtLe38ykrkB4K0goFiZySSCSCHcft27ePjIy4HhGwiFGQVihI5I7e3t6Kigr/Po4nT55kxxGwREFaoSCRC+LxeLDjWF1dzX0cgYygIK1QkDJcMTab8+fPl5eXG2PKysp6enrsn5CoZYhahusgHaAgZVhzZLpEIlFXV1dQUGCMqayszNSOI1HLELUMK+k4QEHKMJVM0dPT4+84FhcXt7S0ZPCII1HLELUMBekABSnDVBIYGxv7+7//++BU1YyvA0DUMkQtQ0E6QEHKMJUEJiYm1q5dG41GOzs7s/H8RC1D1DIUpAMUpAxTSbL+/v7sXeNI1DJELUNBOkBByjCVyBC1DFHLUJAOUJAyTCUyRC1D1DIUpAMUpEx+TiVdXV3ff/+9+IfmZ9ROELUMBekABSmTh1NJR0dHQUHB2rVrxYvG5WHUrhC1DAXpAAUpk4drjkxMTFRWVmb2GsdU5GHUrhC1DCvpOEBBAkBYUZBWKEgACCsK0goFiYwYGxtzPQQAU1GQVihI2Ovt7S0rK/v2229dDwTAYyhIKxQkbIyMjFRXV/u346ivr3c9HACPoSCtUJBIW2tra0lJiTFmyZIlTU1N4lNVAcyLgrRCQSINQ0NDe/bs8W/HsXnz5oGBAdcjAjADCtIKBSkTmivGOjo6otGofx/H999/PwdPzwlN1LmPqGW4DtIBClImBGuOxOPxWCwW7Dhm/D6OmRKCqBcLopZhJR0HKEiZxT6VtLe3l5eXG2MKCgoaGhpy+YjjYo96ESFqGQrSAQpSZvFOJYlEIthxfOaZZ65du+Z6RPNYvFEvOkQtk2ZB+p/Mzma2T2zzcKvz5883Nzc3PO6FF14wxnz44Ye5MMJwbzU00+s7p0Y441b/+q//+pd/+ZfGmL/4i784duzYf/7nf+baCKdv1d/f39fX19/fn7MjDM1WyVHn5ghDs9VQ0gSS+lbG32w2Q7P0bX5uVVdXZ2Zy6tSpHBlhiLcamqkgc2qEU75/bGwseMFs2LDh9OnTuTZCtmKr/NkqeQJJfSv2IFPdqq+v78MPPzz6uMrKSvYgNVsNLbY9yImJic2bNy9btuz3v//9n//85xwc4WxbsQcp24o9SNlWQ+ntQc74U5EijkHKDM1UkDlucHBwMV7juBijXqSIWia9qClIKxSkDFOJDFHLELUMBekABSnDVCJD1DJELUNBOkBBysRzeM2Rnp6ec+fOuR5FxuRy1CFD1DLpRU1BWqEg0dHRYYwpLy/PwUXjANigIK1QkJiYmKisrDx58qTrgQDIMArSCgUJAGFFQVqhIAEgrChIKxRkXhkdHXU9BAA6FKQVCjJ/dHR0FBcX9/b2uh4IABEK0goFmQ9GRkaC23HU1NS4Hg4AEQrSCgUp4+qKsfb29uLiYmNMcXFxS0tLLt/HMVO4OE+GqGW4DtIBClJGv+bI8PDwzp07/R3HrVu3Dg4Oyn60WyzvIkPUMqyk4wAFKSOeStrb20tKSvwdx6ampnzYcQwwa8sQtQwF6QAFKSObSoaGhoIdx+3bt+fPjmOAWVuGqGUoSAcoSBnNVNLZ2ekfcSwpKWlvb8/qz8pZzNoyRC1DQTpAQcpkeyoZHR2trq72dxwrKytHRkay9INyH7O2DFHLUJAOUJAyWZ1KOjs7y8rK/COOra2teXXEcTpmbRmilqEgHaAgZbI0lSQSieQjjkxVHrO2EFHLUJAOUJAy2ZtKtm/fvmTJkjy5xjEVzNoyRC1DQTpAQcpkbyoZGhpihkrGrC1D1DIUpAMUpAxrjsgQtQxRy7CSjgMUJACEFQVphYJcRL7++mvuxQEgdRSkFQpysTh//nxBQUE0GuWejgBSREFaoSAXi4mJicrKSk5VBZA6CtIKBQkAYUVBWqEgASCsKEgrFGQO4qR5ABlBQVqhIGVSvIyppaWlpKSEs1VtcHGeDFHLcB2kAxSkzLwLYYyMjMRiMX9V1VgsJhxa2LC8iwxRy7CSjgMUpMzcr+/Ozs6SkhL/dhwtLS1jY2Pa0YUKs7YMUctQkA5QkDKzvb7j8fiePXv8HcetW7cODg66GF2oMGvLELUMBekABSkz4+s7+T6OXOOYKczaMkQtQ0E6QEHKTHl9j46OBkcct2/fPjw87HR0ocKsLUPUMhSkAxSkTPLru6ury99xLCsra29vZ8cxs5i1ZYhahoJ0gIKU8V/f//7v/15XVxfsOI6MjLgeVwgxa8sQtQwF6QAFKTM0NPSP//iPTzzxhH/E8eTJk+w4ZgmztgxRy1CQDlCQGolEYtu2bZFIxBhTXV3NEcesYtaWIWoZCtIBClLmb/7mb371q1+dPn3a9UDCj+VdZIhahpV0HKAgZUZGRriVIwAlCtIKBQkAYUVBWqEgASCsKEgrFGTGff3119yLA0AuoCCtUJCZ9fXXXxcUFESjUQ43AnCOgrRCQWZcdXV1Q0MD1zgCcI6CtEJBAkBYUZBWKEgZrhiTIWoZopbhOkgHKMi0jY6OLuhzVNYckSFqGaKWYSUdByjI9PT09ESj0e7u7tQ3YSqRIWoZopahIB2gIBdqeHi4urravx3HK6+8kvqGTCUyRC1D1DIUpAMU5IJ0dnb693GMRqM9PT0L2papRIaoZYhahoJ0gIJMUSKRiMVi/o5jLBZL4z6OTCUyRC1D1DIUpAMUZCo6OzvLy8v9+zh2dHSk9yRMJTJELUPUMhSkAxTk3EZHR/fs2ePvOO7cuXNwcDDtp2IqkSFqGaKWoSAdoCDn0NnZWVJS4u84Njc3Wy6Ow1QiQ9QyRC1DQTpAQc4oHo8HRxwrKyuHh4ftn5OpRIaoZYhahoJ0gIKcrru72z9VdcmSJe3t7ZlaVZU1R2SIWoaoZVhJxwEKcoq6urpgx9HmiCMAOEdBWqEgp3jllVdKSkpaW1u5HQeAxY6CtEJBTpFIJNK4xhEAchAFaYWCBICwoiCtUJAAEFYUpJW8LciOjo729nbXowCALKIgreRnQX799df+5f8cbgQQYhSklfwsSM/z6urqGhoalKeqcsWYDFHLELUM10E6kLcFqceaIzJELUPUMqyk4wAFKcNUIkPUMkQtQ0E6EPqCHB0ddT2Eh5hKZIhahqhlKEgHwl2QXV1d5eXlLS0trgfieUwlQkQtQ9QyFKQDYS3I0dHR4HYc1dXVrofjeUwlQkQtQ9QyFKQDoSxIf8fRv5CjpaUlR1ZVZSqRIWoZopahIB0IWUHG4/Gamhp/x3H79u059e+WqUSGqGWIWoaCdCBMBdnR0RGNRo0xJSUlzc3NObLjGGAqkSFqGaKWoSAdCEdBjoyM7Ny5M8fv48hUIkPUMkQtQ0E6EIKCDHYco9FoLt/HkTVHZIhahqhlWEnHgUVdkMk7jtu3b2dhVQBIRkFaWbwFee7cubKyMn/H8eTJkzm74wgArlCQVhZpQdbU1BQUFPg7jsPDw66HAwC5iIK0skgL8pVXXikpKeGGjgAwBwrSyiItyEQiwRFHAJgbBWllkRYkAGBeFKQVChIAwoqCtJLjBdnV1dXU1OR6FJnBFWMyRC1D1DJcB+lALhfk119/7V/jODAw4HosGcCaIzJELUPUMqyk40AuF6Tneb/97W9zeXGcBWEqkSFqGaKWoSAdyPGCDBOmEhmiliFqGQrSAQpShqlEhqhliFqGgnQgRwoyHw7yM5XIELUMUctQkA7kQkF2dXWVl5e3tLQ4HIMAU4kMUcsQtQwF6YDbgozH47FYzD9VtaamxskYZJhKZIhahqhlKEgHHBZkZ2dneXm5MaakpCQ0p6rOgalEhqhliFqGgnTASUEODw9XV1cH93EcHR1V/nRXmEpkiFqGqGUoSAf0BRncx7GkpKSjoyP0O44B1hyRIWoZopZhJR0HlAUZj8fr6uq4jyMAaFCQVmQFef78+aeeesrfcTx58mT+7DgCgCsUpBVBQSYSCXYcAUCPgrSS7YLs7Oz0jzhGo9GOjo4s/RQAwHQUpJWsFmR9fb1/quqePXvYcQQAMQrSSlYL8re//W2+naoKALmDgrSS1YJMJBIjIyPZeGYAwLwoSCu5sBZrnuCKMRmiliFqGa6DdICClGHNERmiliFqGVbScSAjBXnu3LmmpqZMDSmsmEpkiFqGqGUoSAfsC3JoaGjJkiXGmIGBgQwOLHyYSmSIWoaoZShIBzKyB9nU1NTU1MSpqnNjKpEhahmilqEgHeAYpAxTiQxRyxC1DAXpAAUpw1QiQ9QyRC1DQTqwoIIcHBzkc9S0MZXIELUMUctQkA6kXpCtra3FxcUNDQ3ZH1Q4MZXIELUMUctQkA6kUpDDw8NVVVX+qqo7d+6UjS1kmEpkiFqGqGUoSAfmLcj29vbi4mJjTHFxcXNzMx+xpo01R2SIWoaoZVhJx4Gq/zH9SyMjI8GO49atWwcHB/XDAwCk4ueff57+IAVpZbaC7OjoKCkpMcYsWbKEHUcAyGUNDQ2bN2/+6aefpjxOQVqZXpAjIyPbt28Pdhw5ugAAuSwej1dUVBhj9u/f/8svvyR/iYK0MqUg29vby8rK/COOra2t7DgCQO67evXqihUrjDG/+93vkh+nIK0EBTk8PLxnzx5/x7GqquratWuuhwYASNWXX35ZUFBgjGlrawsepCCtBAVZXl7OqaoAsHi1tbUZYwoLC//0pz/5j5jz5883IAV/+7d/W7UQrscLAFiYv/7rvzbGRCKR3/zmN57nmYaGBoPULKggXQ8WAPDQM888U1dX98wzz8z7nU888YQxpqCg4ObNm6avr+8oUlBXV7eggnQ93rCprKw0xlRWVroeSPgRtQxRy5w6daqvr+/UqVNzf9vWrVv9djx8+LDHMUhLQR26Hkj4NXDjFBWiliFqmVSWmjt58qSZcgxSMbTwoiBlmEpkiFqGqGXmLchz585xFmuGUZAyTCUyRC1D1DJzFyTXQWYFBSnDVCJD1DJELTNHQbKSTrZQkDJMJTJELUPUMnPvQR49enTz5s1jY2NTHqcgrVCQMkwlMkQtQ9Qy8x6D5G4emVdXV2eMqaurcz2Q8POvR+rr63M9kPAjahmiluGGyQ7wBhAAch8F6QAFCQC5Lx6PDw0NxePxBW1FQVqhIAEgrChIKxQkAIQVBWkl9YK8e/fuoUOHotFoYWFheXn5q6++evfuXcEIQ+add94xhhdt1jU2Nq5fv76oqGj58uV1dXW3bt1yPaIQun379r59+5YuXbp06dJt27ZdvnzZ9YgwFXONldQLcuPGjVPWjF+zZs3k5KRgkKFx8eLFSCRCQWbbgQMHprxWV65cefPmTdfjCpV79+6VlZUlh1xYWPjNN9+4Hhcew1xjJcWCPHPmjDFm9erVly5d8jzvu+++8/9tnDt3TjLMMPjqq6/8taAoyKzq6+szxuzatevGjRue501OTh4/ftwYU1tb63pooXLkyBFjzI4dO+7fvz8+Pu6/KamsrHQ9LjyGucZKigVZX19vjPnss8+CR/xJ54033sjyAEPirbfeikQizz//PAWZbf7tfn744YfgkcnJSWPMsmXLHI4qfFatWmWMuX37tv+fY2NjxpiioiK3o8IUzDVWUizIbdu2GWOuXbsWPDIwMGCM2bZtW5YHGBLGmIMHD46Pj1OQTvgfALoeRZglEgljTFlZmeuB4DHMNVZSLEj/s8GJiYngEX+uX7FiRZYHGBLBsRkKUu/27dvGmE2bNrkeSGj98MMPtbW1xphjx465HktocR2kAykWZFFR0fRpnU9U0kBB6h09enTKTfKQQadPn/ZPPTty5IjrsYQZK+k4kGJBzjit87FVGihIsYGBgaVLl7744ouuBxJara2tW7ZsWbp0aSQSoSOzh4J0gD1IMQpS6c6dO+Xl5evWrXvw4IHrsYTc2NjYrl27jDHvvfee67GEEwXpQNrHICcmJjgGmQYKUubOnTtr1qypqKi4c+eO67HkBf9Y76pVq1wPJJwoSAdSLMgXX3xxylms169f57KnNFCQGpcuXaqoqFi/fj3tqMSnStlDQTqQYkEePnzYGPPxxx8Hj3z88cfGmPr6+iwPMGwoSIHu7u6ioqItW7bwyWr2RKNRY0zy+kTXrl3zF9hyOKoQoyAdSLEgz54963944q+kc+nSJX8lnTNnzkiGGR4UZLZduXKlsLBw/fr1iUTC9VjCzH/TvH//fn9Nhjt37rz00kvGmMbGRtdDCycK0oHU12Jds2bNlPUtn332WcEIQ4aCzLbdu3ebWbgeWqjcu3evoqJiSsLPP/988pkKyCAK0oHUC/LmzZt1dXUrVqyIRCKlpaUHDx68d++eYIQhw0ydbf4Z1xSkwL179w4dOlRaWhqJRNasWdPY2Dg+Pu56UKFFQTrA/SABIPexko4DFCQAhBUFaYWCBICwoiCtUJAAEFYUpBUKEgDCioK0QkECQFhRkFYoSAAIKwrSCgUJAGFFQVqhIAEg93EdpAMUJADkPlbScYCClEnvDSDSQNQyRC1DQTpAQcqk9/pGGohahqhlKEgHKEgZphIZopYhahkK0gEKUoapRIaoZYhahoJ0gIKUYSqRIWoZopahIB2gIGWYSmSIWoaoZShIByhIGaYSGaKWIWoZCtIBClKGqUSGqGWIWoaCdICClGEqkSFqGaKWYSUdByhIGaYSGaKWIeocR0FaoSBlWHNEhqhliDrHUZBWKEgACCsK0goFCQBhRUFaoSABIKwoSCsUJACEFQVphYIEgLCiIK1QkACQ+7gO0gEKEgByHyvpOEBBynDFmAxRyxC1DAXpAAUpw5ojMkQtQ9QyFKQDFKQMU4kMUcsQtQwF6QAFKcNUIkPUMkQtQ0E6QEHKMJXIELUMUctQkA5QkDJMJTJELUPUMhSkAxSkDFOJDFHLELUMBekABSnDVCJD1DJELUNBOkBByjCVyBC1DFHLsJKOAxSkDFOJDFHLEHWOoyCtUJAyrDkiQ9QyRJ3jKEgrFCQAhBUFaYWCBICwoiCtUJAAEFYUpBUKEgDCioK0QkECQFhRkFYoSADIfVwH6QAFCQC5j5V0HKAgZbhiTIaoZYhahoJ0gIKUYc0RGaKWIWoZCtIBClKGqUSGqGWIWoaCdICClGEqkSFqGaKWSS/q/wdrD9GEWseQHgAAAABJRU5ErkJggg==[/img][/center][b][u]Polar Representation of z[/u][/b][b][br][br]Using trigonometry we have the identification: [/b][math]z=|z|cos(θ)+|z|sin(θ)i=|z|[cos(θ)+sin(θ)i]=|z|cis(θ)[/math] [b]where[/b] [math]a=|z|cos(θ),b=|z|sin(θ)[/math][b].[br][br]The angle [math]θ[/math][/b][b] determined by [/b][math]z[/math][b] can be measured in degrees or [br]radians and restricted to be in a specific interval. [br][br]For example, [math]θ∈[0,2\pi)[/math][/b][b] or [math]θ∈(−\pi,\pi][/math][/b][b].[br][br]Thus the angle can be considered a function of [math]z[/math][/b][b], called [u]the argument of [math]z[/math][/u][/b][b]: [math]Arg(z)=θ[/math][/b][b].[center][br][img width=356,height=199]data:image/png;base64,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[/img][/center][/b]
[b]Complex Numbers in GeoGebra[/b][br][br][b]From the Toolbar:[/b][br] From the point dropdown, select [icon]/images/ggb/toolbar/mode_complexnumber.png[/icon] complex number. Then click on any point in the Graphics Frame to create a complex number. [br][br][b]Comments: [/b]The point will appear in the frame labeled z[sub]1[/sub]. [br]In the Algebra Frame will appear z[sub]1[/sub] = ... + ....i. [br]This number/point is "free" and can be changed with the mouse (reset to the pointer [icon]/images/ggb/toolbar/mode_move.png[/icon]) by moving it, or by entry of a new value by double clicking on current value in the Algebra Frame.[br][br][b]From the Input Bar:[/b][br]Enter the complex number with or without a name: e.g., z1= 2+3i or just 2+3i.[br][br][b]Comments: [/b]A point will appear in the Graphics Frame labeled z1. [br]In the Algebra Frame will appear z1 = 2 + 3i. [br]This number/point is "free" and can be changed with the mouse (reset to the pointer [icon]https://www.geogebra.org/images/ggb/toolbar/mode_move.png[/icon]) by moving it, or by entry of a new value by double clicking on current value in the Algebra Frame.[br][br][b]The Norm of a Complex Number[/b][br][b]From the Input Bar:[/b][br][b]As a function: [/b] Enter abs(...) and hit return. A real number will appear in the Algebra Frame determined as the norm of the complex number. E.g. abs(3+4i) gives ...=5. This real number is dependent on the argument of abs.[br][b]With bar notation:[/b] Enter |...| and hit return. A real number will appear in the Algebra Frame determined as the norm of the complex number. E.g. |3+4i| gives ...=5. This real number is dependent on the number between the bars.[br][br][b]Polar Representation of a Complex Number[/b][br][b]From the Input Bar:[/b][br][b]As a function: [/b] Enter ToPolar(...) and hit return. A point will appear in the Algebra Frame determined with the polar coordinates of the complex number. E.g. ToPolar(3+4i) gives ...=(5; 53.13[math]^\circ[/math]). This point is [br]dependent on the on the argument of ToPolar.[br][b][br]Comment: [/b]The Argument of a complex number is the second coordinate of the Polar Form.[br]This can be found directly by using the "Angle" function applied to the complex number. E.g. Angle(3 + 4ί) gives ...=53.13[math]^\circ[/math].
– Martin Flashman

1.2.

–

1.3.

–

1.4.

–

1.5.

–

2.

Linear and Quadratic Functions and Equations

  • 1. Table and Mapping Diagram for A Complex Linear Function
  • 2. Complex Linear Mapping Diagram II: f(z) = Az
  • 3. Complex Linear Mapping Diagram III (lines)
  • 4. Complex Linear Mapping Diagrams I (Circles)
  • 5. Complex Mapping Diagrams for Quadratic Function and Solving Real Quadratic Equations
  • 6. Mapping Diagrams for Complex Quadratic Functions & Equations
  • 7. Mapping Diagram Solving A Quadratic Equation(Real and Complex)
  • 8. Complex Variable Function Mapping Diagrams and Solving Real Cubic Equations

2.1.

Table and Mapping Diagram for A Complex Linear Function

First Example of a 3D Mapping Diagram for a Complex Linear Function
Consider the complex linear function: [math]f(z)=a+bz[/math] where [i]a [/i]and [i]b[/i] are complex numbers managed as points in the complex plane. [br][br]You can click on the points to change the complex values for [i]a[/i] and [i]b[/i] by moving the points in the plane.[br]The two tables are created as a spread sheet with the entries in the [math]z=a+bi[/math] table indicating the complex numbers in the lattice of the domain used to determine the [i][math]w=f(z)[/math] [/i]table determined as complex numbers using arithmetic. [br]In this example no complex "function" has been created.[br]Check the box to see the mapping diagram that corresponds to the data in the table. [br]The 3 dimensional mapping diagram is created using the data in the spread sheet.[br]The domain plane is parallel to the target plane and the arrows go from points/complex numbers [math]z[/math] in the domain to the corresponding points/complex numbers [math]w=f(z)[/math] in the target plane.
– Martin Flashman

2.2.

–

2.3.

–

2.4.

–

2.5.

–

2.6.

–

2.7.

–

2.8.

–

3.

Mapping Diagrams to Visualize Elementary Functions

  • 1. Mapping Diagrams: Complex Analysis -functions with complex parameters. circles
  • 2. Mapping Diagrams: Complex Analysis-functions with lines or circles
  • 3. Mapping Diagram: Complex Analysis. Realign Target.
  • 4. Complex Linear Mapping Diagram f(z) = Bz; parametric curve
  • 5. Moebius Transformation: MD circle with spherical projection
  • 6. Complex Mapping Diagram for Moebius Function (circles)
  • 7. Mapping diagram for Moebius Function
  • 8. Mapping diagram for Moebius Transformation: roots and poles, lines and circles
  • 9. Mapping Diagram for Complex Power Function (Circles)
  • 10. Mapping Diagram for z^n with Curve on Riemann Surface
  • 11. Mapping Diagram for z^n
  • 12. Complex Mapping diagram with homotopy
  • 13. Mapping Diagram Powers Homotopy
  • 14. Mapping Diagrams for Quartic Equations
  • 15. Mapping Diagrams for Visualizing Roots for Polynomials
  • 16. Complex plane transformations: mapping a line
  • 17. Complex square mapping between two planes

3.1.

Mapping Diagrams: Complex Analysis -functions with complex parameters. circles

A simple but still interesting example is presented with the complex function [math]g(z)=z^2+az+b[/math].[br]In this GeoGebra activity, The parameters [math]a[/math] and [math]b[/math] are controlled by the complex numbers (points) [b]a[/b] and [b]b[/b] in the domain frame. A circle of radius [math]\delta[/math] controlled by the slider and center at the complex number (point) [math]z_#[/math] is used to locate [b]m[/b] points n the circle where [b]m[/b] is con trolled by another slider. [br][br]The 3-dimensional frame has the mapping diagram for the function visualized using the points on the circles in the domain frame along with the point [math]z_#[/math]. For comparison the graph of an associated real quadratic function is shown in the third 2-dimensional frame.[br][br]Changes can be made by moving [math]z_#[/math], [b]m[/b], [b]a, [/b]b or [math]\delta[/math], or by entering a new complex function using parameters [b]a[/b] or [b]b[/b]. This example can be removed from sight in the activity by unchecking the box labelled "Show function g with parameters a and b".[br][br]A more subtle example can be shown by checking the box labelled [br]"Show function h with parameters a, b, c and d".[br]This will show the function [math]h(z)=\frac{az+b}{cz+d}[/math] using parameters [b]a,b,c,[/b] and [b]d[/b] controlled by the appropriate complex numbers (points) in the domain frame. The other two frames are as with the quadratic example.[br]Again the diagram can be controlled by changing the parameters. [br]
– Martin Flashman

3.2.

–

3.3.

–

3.4.

–

3.5.

–

3.6.

–

3.7.

–

3.8.

–

3.9.

–

3.10.

–

3.11.

–

3.12.

–

3.13.

–

3.14.

–

3.15.

–

3.16.

–

3.17.

–

4.

Limits and Continuity for Complex Functions

  • 1. Mapping Diagrams: Complex Analysis- functions and continuity
  • 2. Limits for Complex Functions
  • 3. Mapping Diagram:Limits for Complex Functions
  • 4. Limits for Complex Functions
  • 5. Limits for Complex Functions
  • 6. Continuity and Open Sets (Work in Progress)

4.1.

Mapping Diagrams: Complex Analysis- functions and continuity

[b][center]Continuity at a Point[/center]Definition: [/b]Suppose [math]U\subset\mathbb{C}[/math] is an open set and [math]f:U\to\mathbb{C}[/math] with [math]z_0\in U[/math]. [br]We say [math]f[/math] is continuous at [math]z_0[/math] if [math]\lim_{z\to z_0}f(z)=f(z_0)[/math].
– Martin Flashman

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

5.

Mapping Diagrams to Visualize Differentiation and Applications

  • 1. Mapping Diagram Visualizing Complex Derivative:Powers
  • 2. Mapping Diagram Visualizing Complex Differential
  • 3. Mapping Diagram for Newton's Method in Complex Analysis

5.1.

Mapping Diagram Visualizing Complex Derivative:Powers

Mapping Diagram Visualizing Complex Derivative for Powers
[b][center]The Complex Derivative and the Differential[/center]Suppose [math]U\subset\mathbb{C}[/math] is a domain, [math]z_0\in U[/math], and [math]f:U\longrightarrow\mathbb{C}[/math].[br][br]Definition: [/b]The [b]derivative[/b] of [math]f[/math] at [math]z_0[/math], denoted [math]f'(z_0)[/math], is defined by[br] [math]f'(z_0)\equiv\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}[/math] [b]provided the limit exists.[br][/b]If [math]f'(z)[/math] exists for all [math]z[/math] in some open set containing [math]z_0[/math] , [math]f[/math] is called [b]holomorphic[/b] at [math]z_0[/math].[br][br][b]Definition: [/b]Suppose [math]f[/math] is holomorphic at [math]z_0[/math].[br]The [b]differential[/b] of [math]f[/math] at [math]z_0[/math] for [math] \Delta z\in\mathbb{C}[/math] , denoted [math]df(z_0,\Delta z)[/math], is defined by [math]df(z_0,\Delta z)=f'(z_0)\Delta z[/math] .[br][br][b]Fact: [/b] If [math]f[/math] is holomorphic at [math]z_0[/math] and [math]z_0+\Delta z\in U[/math], then [math]f(z_0+\Delta z)\approx f(z_0)+df(z_0,\Delta z)[/math].
– Martin Flashman

5.2.

–

5.3.

–

6.

Mapping Diagrams to Visualize Integration

  • 1. Complex Line Integral: Definition and F.T. Examples
  • 2. Mapping Diagram for Cauchy Integral Formula
  • 3. Complex Integration from list
  • 4. Integral Examples from List
  • 5. Mapping Diagram: Complex Integration Parametric Function
  • 6. Mapping Diagram for Complex Integration over A Circle
  • 7. Mapping Diagram: Complex Integration of 1/z
  • 8. Complex Integration: z^2 over line segment
  • 9. Complex Integration: z^2 over curve
  • 10. The Winding Number of a Curve about a Point

6.1.

Complex Line Integral: Definition and F.T. Examples

Definition of the integral for complex analysis and its mapping diagram:[br]Suppose : [math]f:\mathbb{C}\to\mathbb{C}[/math] and [math]\gamma:[a,b]\to\mathbb{C}[/math][br][b]The definite integral of[/b] [math]f[/math][br][math]\int_γf(z)dz\ =\lim_{n \to \infty}(\sum_{k=0}^{n-1}f(z_k)\Delta z_k)[/math] [br]where [math]Δz_k = z_{k+1}-z_k; z_k = γ(s_k); s_k =t_0+ kΔt; Δt=\frac{t_1-t_0}n[/math].[br]
– Martin Flashman

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

7.

Series: Taylor and Laurent

  • 1. Taylor Polynomials for Complex Functions: Mapping Diagrams

7.1.

Taylor Polynomials for Complex Functions: Mapping Diagrams

[b]Taylor's theorem[/b] generalizes to functions[math]f:U\longrightarrow\mathbb{C}[/math] which are [url=https://en.wikipedia.org/wiki/Complex_differentiable]complex differentiable[/url] in an open subset [math]U⊂\mathbb{C}[/math] of the [url=https://en.wikipedia.org/wiki/Complex_plane]complex plane[/url].[br][br]The [b]Taylor polynomial[/b] holds in the form similar to that for real analysis: [br][math]f(z)=P_k(z)+R_k(z)[/math] , [math]P_k(z)=\sum_{j=0}^k \frac {f^{(j)}(c)}{j!}(z-c)^j[/math] , and [math]R_k(z)=f(z)-P_k(z)[/math],[br]which the following figure visualizes and compares using mapping diagrams restricted to circles.
Taylor Polynomial in GeoGebra
– Martin Flashman

Information