Complex mappings
Instructions:
Consider f(z) = u(x,y) + i v(x,y).
First find the real and imaginary components of the function f(z). Then type those values in the corresponding input boxes u and v.
Drag the points A and B to change the size of the square, or drag the square to change its position.
Example:
f(z) = z2 = x2 - y2 + i (2xy), then u = x^2 - y^2 and v = 2xy
More examples:
1. f(z) = exp(z) = exp(x)cos(y) +i exp(x)sin(y), then u = exp(x)cos(y) and v = exp(x)sin(y)
2. f(z) = sin(z) = sin(x)cosh(y) + i cos(x)sinh(y), then u = sin(x)cosh(y) and v = cos(x)sinh(y)
For z = r exp(i ), with r = and = Arg(z), that is, ; we need to use the function atan2. For example:
1. Consider the multiple value function f(z) = z1/2 = r1/2 exp( i (1/2 ) ). Then
f(z) = (x^2 + y^2)^(1/4)cos( 1/2atan2(y, x) ) + i (x^2 + y^2)^(1/4)sin(1/2 atan2(y, x))
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))
The second value of z1/2 is given by
f(z) = (x^2 + y^2)^(1/4)cos(1/2atan2(y, x) + pi) + i (x^2 + y^2)^(1/4)sin(1/2atan2(y, x) + pi)
2. Now consider the function f(z) = log(z) = log( ( x^2 + y^2 )^(1/2) ) + i atan2(y, x), then we have that
u = log((x^2 + y^2)^(1/2)) and v = atan2(y, x)