A **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 = z ^{2}** but recall z = x + iy

** w = (x + iy) ^{2} = (x + iy)(x + iy) = x^{2} + ixy + iyx -y^{2}**

** = (x ^{2} – y^{2}) + i(2xy)**

In polar form** z = re**^{iθ }

**w = z ^{2 }**

**= (re**

= r

^{iθ}) = r^{2}(e^{iθ})= r

^{2}e^{i2θ}Example: ii) **w = z ^{1/2}** Two valued function

**z = re ^{iθ} ** since z = 1 + i, and r = √(x

^{2}+ y

^{2})

**w = (1 + i) ^{1/2}**

=

**√2 e**for k = 0, 1, 2, …

^{iӅ/4 }**w = (1 + i)**^{1/2 }principal value = (√2 e^{iӅ/4})^{1/2}

for 0 ≤ θ ≤ 2Ӆ^{c }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}, z^{1/2} = (√2 e ^{i9/4Ӆ})^{1/2} = 2^{1/4} e^{i9/8Ӆ}**

**Elementary Functions**

for polynomials

**w = a _{0} z^{n} + a_{1} z^{n-1} + … + a_{n-1} z, ** then tan = P(z)

where a_{0} ≠ 0, a_{1}, …, a_{n} 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 = e ^{z }= e ^{x + iy} = e^{x}. e^{y} = e^{x} (cos y + i sin y)**

Example: e^{2 + 3/4Ӆi} = e^{2}_{. }e^{3/4Ӆi}

^{ } ** = e ^{2}(cos ¾ Ӆ) + isin ¾ Ӆ**

** = e ^{2}(-^{√2}/_{2 }+ i ^{√2}/_{2})**

** = -e ^{2} ^{√2}/_{2 }+ i ^{√2}/_{2}**

4. **Trigonometry functions**

cos z = ^{1}/_{2}(e^{iz} + e^{-iz}), sin z = ^{1}/_{2i}(e^{iz} + e^{-iz})

5. **Trigonometry identity and Hyperbolic identities**

cos^{2} + sin^{2} = 1 and cosh^{2} – sinh^{2} = 1

6. **Logarithmic function** (Natural base, log_{e} = ln)