Consider f(z) = u(x,y) + [i][b]i [/b][/i]v(x,y).[br][br]First find the real and imaginary components of the function f(z). Then type those values in the corresponding input boxes u and v.[br][br]Drag the points A and B to change the size of the square, or drag the square to change its position.

f(z) = z[sup]2[/sup] = x[sup]2[/sup] - y[sup]2[/sup] + [i][b]i [/b][/i](2xy), then u = x^2 - y^2 and v = 2xy

1. f(z) = exp(z) = exp(x)cos(y) +[i][b]i[/b] [/i]exp(x)sin(y), then u = exp(x)cos(y) and v = exp(x)sin(y)[br][br]2. f(z) = sin(z) = sin(x)cosh(y) +[i] [b]i[/b] [/i]cos(x)sinh(y), then u = sin(x)cosh(y) and v = cos(x)sinh(y)[br][br]For z = r exp([i]i[/i] [math]\Theta[/math]), with r = [math]\sqrt{x^2+y^2}[/math] and [math]\Theta[/math]= Arg(z), that is, [math]-\pi<\Theta\le\pi[/math]; we need to use the function [url=https://en.wikipedia.org/wiki/Atan2]atan2[/url]. For example:[br][br]1. Consider the multiple value function f(z) = z[sup]1/2[/sup] = r[sup]1/2[/sup] exp( [b][i]i[/i][/b] (1/2 [math]\Theta[/math]) ). Then[br][br] f(z) = (x^2 + y^2)^(1/4)cos( 1/2atan2(y, x) ) + [i][b]i[/b][/i] (x^2 + y^2)^(1/4)sin(1/2 atan2(y, x))[br][br]Thus u = (x^2 + y^2)^(1/4)cos(1/2 atan2(y, x)) and v = (x^2 + y^2)^(1/4)sin(1/2 atan2(y, x))[br][br]The second value of z[sup]1/2[/sup] is given by[br][br] f(z) = (x^2 + y^2)^(1/4)cos(1/2atan2(y, x) + pi) + [i][b]i[/b][/i] (x^2 + y^2)^(1/4)sin(1/2atan2(y, x) + pi)[br][br]2. Now consider the function f(z) = log(z) = log( ( x^2 + y^2 )^(1/2) ) + [i][b]i [/b][/i]atan2(y, x), then we have that [br] [br]u = log((x^2 + y^2)^(1/2)) and v = atan2(y, x)