Function of a Complex Variables

complex function of a function of a complex variables is a function whose domain and range are subsets of the complex plane. This is also expressed by saying that the independent variable and the dependent variable both are complex numbers. If f(x) is analytic everywhere in the complex plane, it is called entire function. To distinguish analytic functions from generic complex-valued functions of complex variable, we use the notation f(x) for the former and w(x,x*) .

Functions of a Complex Variables

If to each value which a complex number z can take their corresponding one or more values of a complex number w, we say w is a function of z which is written as w = f(z). w can be single valued or many – valued.

A many valued functions, each number or which is called a branch of function.

We usually choose the branch as a principal branch and the value of function corresponding to their branch is called the principal value.

Example: i) w = z2  but recall z = x + iy

w = (x + iy)2 = (x + iy)(x + iy) = x2 + ixy + iyx -y2

= (x2 – y2) + i(2xy)

In polar form  z = reiθ

w = z = (re)  = r2 (e)
= r2 ei2θ

Example: ii) w = z1/2 Two valued function

z = re          since z = 1 + i, and  r = √(x2 + y2)

w = (1 + i)1/2
= √2 eiӅ/4   for k = 0, 1, 2, …

w = (1 + i)1/2  principal value = (√2 eiӅ/4)1/2

for 0 ≤ θ ≤ 2Ӆc General argument

Ӆ/4 + 2Ӆk   k = ±1, ±2, ±3, …

z = √2 e i(Ӆ/4+ 2Ӆ) = √2 e i9/4Ӆ

z = √2 e i9/4Ӆ, z1/2 = (√2 e i9/4Ӆ)1/2 = 21/4 ei9/8Ӆ

1. Elementary Functions

for polynomials

w = a0 zn + a1 zn-1 + … + an-1 z,           then tan = P(z)

where a0 ≠ 0, a1, …, an are complex constant.

Example: w = az + b (linear polynomial / transformation) where degree is 1

2. Partial algebraic function

where  w = P(z)/Q(z) where P and Q are polynomials

Examples: w = (az + b)/ (cz + d)     (Bi linear transformation)

3. Exponential Function

w = ez = e x + iy = ex. ey = ex (cos y + i sin y)

Example: e2 + 3/4Ӆi = e2. e3/4Ӆi

= e2(cos ¾ Ӆ) + isin ¾ Ӆ

= e2(-√2/2 + i √2/2)

= -e2 √2/2 + i √2/2

4. Trigonometry functions

cos z = 1/2(eiz + e-iz),           sin z = 1/2i(eiz + e-iz)

5. Trigonometry identity and Hyperbolic identities

cos2 + sin2 = 1 and cosh2 – sinh2 = 1

6. Logarithmic function (Natural base, loge = ln)