Complex Functions -- A Dip Into Differentiability

Dorothea Nguyen, Mar 7, 2017

In this book, we will look into complex functions and introduce a little bit on what it means to be differentiable in the complex plane. The applets will allow for a better geometric understanding of complex numbers and operations performed on them.

Table of Contents

  1. Complex Number Primer
    1. Complex Number Addition
    2. Complex Number Multiplication
  2. Some Helpful Definitions
    1. Complex Numbers
    2. Complex Field
    3. Complex Functions
  3. Complex Functions
    1. Function Grapher -- Complex Inputs
    2. Complex Functions -- Conjugates
    3. More Functions
  4. Differentiation
    1. Setting Up The Magic
    2. Derivative
    3. Derivatives of a Complex Function
    4. Limits for Complex Functions
  5. What is next?
    1. Applications

1.

Complex Number Primer

Before we begin, let's tackle some basic operations that can be done on complex numbers. The definition of these basic operations differ from their analogue in the real numbers, so let's attempt to represent these operations visually in the plane. If you're already familiar with these operations, feel free to skip to the second chapter!

  • 1. Complex Number Addition
  • 2. Complex Number Multiplication

1.1.

Complex Number Addition

Here we have the addition of two complex numbers. The two points [math]z_1[/math] and [math]z_2[/math] can be moved around.[br]The third point, [math]z_3[/math] , represents the sum [math]z_1+z_2[/math].[br][br]Can you find a geometric representation of the addition of two complex numbers?
Modified slightly from original app, found on GeoGebra Materials Team[br][br][i][url=https://www.geogebra.org/geogebra+team]https://www.geogebra.org/geogebra+team[/url][/i]
– Dorothea Nguyen

1.2.

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2.

Some Helpful Definitions

Browse each of these sections as needed. They are not necessary if you already have a foundation with Complex Analysis.

  • 1. Complex Numbers
  • 2. Complex Field
  • 3. Complex Functions

2.1.

Complex Numbers

Complex Number
We can think of complex numbers as pairs of real numbers with the form:[br] [img]https://latex.codecogs.com/gif.latex?%5Cmathbb%7BC%7D%3D%7B%7D%7B%28x%2Cy%29%3Ax%2Cy%5Cepsilon%20%5Cmathbb%7BR%7D%7D[/img][br]Where addition is defined as:[br] for [img]https://latex.codecogs.com/gif.latex?x%2Cy%2Ca%2Cb%5Cepsilon%20%5Cmathbb%7BR%7D[/img][br] [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29+%28a%2Cb%29%3D%28x+a%2Cy+b%29[/img][br]And multiplication defined as:[br] [img]https://latex.codecogs.com/gif.latex?%28x%2Cy%29%28a%2Cb%29%3D%28xa-yb%2Cxb+ya%29[/img][br][br]
– Mason

2.2.

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2.3.

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3.

Complex Functions

Now that we have had a (very) brief explanation on how complex numbers operate, let us introduce complex functions.

  • 1. Function Grapher -- Complex Inputs
  • 2. Complex Functions -- Conjugates
  • 3. More Functions

3.1.

Function Grapher -- Complex Inputs

This applet allows us to plug in a complex number as an input to a function.[br]Move [math]z_1[/math] around and observe the output [math]z_2[/math]. Turn on the trace for [math]z_2[/math] to track its movement as [math]z_1[/math] changes.[br]If you'd like, try it again on a different function in terms of [math]z_1[/math][br]How does this differ from functions in the real plane?
Modified from[i] [url=https://www.geogebra.org/material/show/id/136234]https://www.geogebra.org/material/show/id/136234[/url][/i]
– Dorothea Nguyen

3.2.

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3.3.

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4.

Differentiation

  • 1. Setting Up The Magic
  • 2. Derivative
  • 3. Derivatives of a Complex Function
  • 4. Limits for Complex Functions

4.1.

Setting Up The Magic

The previous applet was trying to show how the limit of some function ([math]z_1z_2[/math], where the two are conjugates) can take different values depending on [i]how[/i] we approached [math]z=0[/math]. [br][br]This is not entirely different from the definition of limits in the real plane. The most noticeable difference is that within the reals, we can approach a point [math]p[/math] from only two directions. However, in the previous applet we saw that it is possible to approach [math]z=0[/math] from many directions, say from [math]i[/math] or [math]-i[/math] on some kind of curve we create. Let us now introduce derivatives of complex functions.
– Dorothea Nguyen

4.2.

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4.3.

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4.4.

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5.

What is next?

  • 1. Applications

5.1.

Applications

If you are interested in why these concepts are so widely used, read the article through the following link (You may have to copy and paste the link into an address bar):[br][br][i][url=https://www.quora.com/If-complex-numbers-are-imaginary-then-what-is-their-significance-in-the-real-world-Why-should-we-study-complex-numbers]https://www.quora.com/If-complex-numbers-are-imaginary-then-what-is-their-significance-in-the-real-world-Why-should-we-study-complex-numbers[/url][/i][br][br]One of the main reasons for the study of complex numbers is for using the Residue Theorem, which is essentially a method for computing the area, or residue, of unbounded functions around the point(s) at which the function is unbounded. Differentiation plays a key role in establishing the criterion for applying the Residue Theorem, but that's hardly a topic for a Monday![br][br]Phew, hope you enjoyed the read![br]
– Dorothea Nguyen, Mason

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