Complex Function Mapping Theory and Examples (GATE ) For the function of a complex variable w = ln z

Complex Function Mapping

A complex valued function f(z) of complex variable is called complex function. It maps one complex number to another complex number and it is often written as w = f(z) where z is a complex number and w = f(z) is value of the complex function at z. Lets take an example
w=f(z)=z^^2+4

So a complex function can be expressed or written in two different ways. As a function of z or in terms of the real and imaginary parts of w, ie, u and v, which are functions of x and y, ie, the real and imaginary parts of complex number z.

It is not possible to plot graph of complex functions like we do for real functions. Because complex functions need 4 dimensions to visualize the mapping. Two dimensions for z, i.e. x and y, and two dimensions for w, i.e. u and v. So, to visualize complex functions we use two planes: a z-plane and a w-plane. (Domain in z-plane, Set of images in w-plane)

Complex function mappings are often specified in terms of condition on the complex function or real or imaginary parts of the complex function.

Example 1 For the function of a complex variable w = ln z (where w = u + j v and z = x + j y ) the u = constant Lines mapped in the z-plane as
(a) Set of radial straight lines (b) set of concentric circles
(c) set of confocal hyperbolas (d) set of confocal elliplses


Example 2
Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S= {z : |z| = 1}. Consider the function f(z)= zz* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane
(a) unit circle (b) horizontal axis line segment from origin to (1, 0)
(c) the point (1, 0) (d) the entire horizontal axis

Example 3
A point z has been plotted in the complex plane as shown in the figure. The plot of complex number y = 1/z is


Example 4
GATE 2002 (IN): The bilinear transformation w=\frac{z-1}{z+1}
(a) Maps the inside of the unit circle in the z-plane to the left half of the w-plane
(b) Maps the outside of the unit circle in the z-plane to the left half of the w-plane
(c) Maps the inside of the unit circle in the z-plane to the right half of the w-plane
(d) Maps the outside of the unit circle in the z-plane to the right half of the w-plane



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