Introduction to Elementary Complex Functions

If for each value of the complex variable $z = (x + iy)$ in a given region R, we have one or more values of $w (= u + iv)$, then $w$ is said to be a complex function of z and we write $w = u(x, y) + iv(x, y) = f(z)$ where $u, v$ are real functions of x and y.

If to each value of z, there corresponds one and only one value of w, then w is said to be a single-valued function of z otherwise a multi-valued function. For example, $w = \frac{1}{z}$ is a single-valued function and $w = \sqrt{z}$ is a multi-valued function of z.

The former is defined at all points of the z-plane except at z = 0 and the latter assumes two values for each value of z except at z = 0.

Exponential Function

Exponential Function:

When x is real, we are already familiar with the exponential function

$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \dots + \frac{x^n}{n!} + \dots \infty.$

Similarly, we define the exponential function of the complex variable $z = x + iy$, as

$e^z \text{ or } \exp(z) = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \dots + \frac{z^n}{n!} + \dots \infty. \quad \quad \text{(i)}$

Properties:

1. Exponential form of $z = re^{i \theta}$

   Putting x = 0 in (i), we get,

   $e^{iy} = 1 + \frac{iy}{1!} + \frac{(iy)^2}{2!} + \frac{(iy)^3}{3!} + \frac{(iy)^4}{4!} + \dots \infty$

   $= \left( 1 - \frac{y^2}{2!} + \frac{y^4}{4!} - \dots \right) + i \left( y - \frac{y^3}{3!} + \frac{y^5}{5!} - \dots \right) = \cos y + i \sin y$

   Thus,

   $e^z = e^x \cdot e^{iy} = e^x (\cos y + i \sin y)$

   Also,

   $x + iy = r(\cos \theta + i \sin \theta) = re^{i \theta}. \text{ Thus, } z = re^{i \theta}$

2. $e^z$ is periodic function having imaginary period $2 \pi i$; $[e^{z + 2n \pi i} = e^z, \; e^{2n \pi i} = e^0 = e^x]$

3. $e^z$ is not zero for any value of z.

   Since,

   $e^z = e^x \cdot e^{i y} = re^{i \theta} \text{ or } e^x \cdot e^{i y} = re^{i \theta}$

   Therefore,

   $r = e^x > 0, \; y = \theta, \; |e^{iy}| = 1,$

   $\text{Thus } |e^z| = |e^x|, \; |e^{iy}| = e^x \neq 0.$

4. $e^{\overline{z}} = \overline{e^z}$

   Since,

   $e^{\overline{z}} = e^{x - iy} = e^x \cdot e^{-iy} = e^x (\cos y - i \sin y)$

   $= e^x (\cos y + i \sin y) = e^z$

Trignometric and Hyperbolic Function

Trigonometric Functions

Trigonometric functions relate angles in a right triangle to the ratios of the triangle's side lengths. They are periodic functions and are fundamental in the study of waves, oscillations, and circular motion.

1. Sine ($\sin \theta$): Ratio of the length of the opposite side to the hypotenuse.

   $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

2. Cosine ($\cos \theta$): Ratio of the length of the adjacent side to the hypotenuse.

   $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

3. Tangent ($\tan \theta$): Ratio of the length of the opposite side to the adjacent side.

   $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}}$

4. Cosecant ($\csc \theta$): Reciprocal of sine.

   $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$

5. Secant ($\sec \theta$): Reciprocal of cosine.

   $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$

6. Cotangent ($\cot \theta$): Reciprocal of tangent.

   $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} = \frac{\text{adjacent}}{\text{opposite}}$

Key Trigonometric Identities:

• Pythagorean Identity:

  $\sin^2 \theta + \cos^2 \theta = 1$

• Angle Sum and Difference Formulas:

  $\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$

  $\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$

  $\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

• Double Angle Formulas:

  $\sin 2\theta = 2 \sin \theta \cos \theta$

  $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta$

  $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

Hyperbolic Function

Hyperbolic Function:

If x be real or complex,

(i) $\frac{e^x - e^{-x}}{2}$ is defined as hyperbolic sine of x and is written as $\sinh x$.

$\sinh x = \frac{e^x - e^{-x}}{2}$

(ii) $\frac{e^x + e^{-x}}{2}$ is defined as hyperbolic cosine of x and is written as $\cosh x$.

$\cosh x = \frac{e^x + e^{-x}}{2}$

Thus, $\sinh x = \frac{e^x - e^{-x}}{2}$ and $\cosh x = \frac{e^x + e^{-x}}{2}$.

Also we define,

$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}, \quad \coth x = \frac{1}{\tanh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$

$\operatorname{sech} x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}, \quad \operatorname{cosech} x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$

Properties:

• Periodic functions: $\sinh z$ and $\cosh z$ are periodic functions having imaginary period $2\pi i$.

  $\sinh(z + 2\pi i) = \sinh z ; \quad \cosh(z + 2\pi i) = \cosh z$

• Even and odd functions: $\cosh z$ is an even function while $\sinh z$ is an odd function.

• $\sinh 0 = 0$, $\cosh 0 = 1$, $\tanh 0 = 0$.

• Relations between hyperbolic and circular functions.

  Since for all values of \theta ,

  $\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \quad \text{and} \quad \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$

  Therefore, putting $\theta = ix$, we have

  $\sin ix = \frac{e^{-x} - e^x}{2i} = \frac{-(e^x - e^{-x})}{2i} = i \sinh x$

  $\cos ix = \frac{e^{-x} + e^x}{2} = \cosh x$

  $\sinh ix = i \sin x \quad \text{... (i)}$

  $\cosh ix = \cos x \quad \text{... (ii)}$

  $\tanh ix = i \tan x \quad \text{... (iii)}$

  $\sinh ix = i \sin x \quad \text{... (iv)}$

  $\cosh ix = \cos x \quad \text{... (v)}$

  $\tanh ix = i \tan x \quad \text{... (vi)}$

Inverse Trignometric and Hyperbolic

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the basic trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. They are used to find angles when the trigonometric values are known.

Definitions:

• If $y = \sin^{-1} x$, then $\sin y = x$ for $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

• If $y = \cos^{-1} x$, then $\cos y = x$ for $0 \leq y \leq \pi$

• If $y = \tan^{-1} x$, then $\tan y = x$ for $-\frac{\pi}{2} < y < \frac{\pi}{2}$

• If $y = \csc^{-1} x$, then $\csc y = x$ for $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}, y \neq 0$

• If $y = \sec^{-1} x$, then $\sec y = x$ for $0 \leq y \leq \pi, y \neq \frac{\pi}{2}$

• If $y = \cot^{-1} x$, then $\cot y = x$ for $0 < y < \pi$

Principal Values:

The principal values for inverse trigonometric functions are as follows:

$\sin^{-1} x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

$\cos^{-1} x \in \left[0, \pi\right]$

$\tan^{-1} x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

$\csc^{-1} x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \text{ where } y \neq 0$

$\sec^{-1} x \in \left[0, \pi\right] \text{ where } y \neq \frac{\pi}{2}$

$\cot^{-1} x \in \left(0, \pi\right)$

Properties:

• Reciprocal Identities:

  $\sin^{-1} x = \csc^{-1} \frac{1}{x}, \quad \cos^{-1} x = \sec^{-1} \frac{1}{x}, \quad \tan^{-1} x = \cot^{-1} \frac{1}{x}$

• Sum and Difference Formulas:

  $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$

  $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$

  $\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}$

• Negative Argument Identities:

  $\sin^{-1}(-x) = -\sin^{-1}(x)$

  $\cos^{-1}(-x) = \pi - \cos^{-1}(x)$

  $\tan^{-1}(-x) = -\tan^{-1}(x)$

• Double Angle Formulas: For values $|x| < 1$,

  $2 \sin^{-1} x = \sin^{-1} (2x \sqrt{1 - x^2})$

  $2 \tan^{-1} x = \tan^{-1} \left(\frac{2x}{1 - x^2}\right)$

• Derivatives:

  $\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1$

  $\frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1$

  $\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}, \quad x \in \mathbb{R}$

  $\frac{d}{dx} \csc^{-1} x = -\frac{1}{|x|\sqrt{x^2 - 1}}, \quad |x| > 1$

  $\frac{d}{dx} \sec^{-1} x = \frac{1}{|x|\sqrt{x^2 - 1}}, \quad |x| > 1$

  $\frac{d}{dx} \cot^{-1} x = -\frac{1}{1 + x^2}, \quad x \in \mathbb{R}$

Inverse Hyperbolic Function

Inverse Hyperbolic Function:

If $\sinh u = z$, then $u$ is called the hyperbolic sine inverse of $z$ and is written as $u = \sinh^{-1} z$. Similarly, we define $\cosh^{-1} z$, $\tanh^{-1} z$, etc.

The inverse hyperbolic functions, like other inverse functions, are many-valued, but we shall consider only their principal values.

• To show that

(i) $\sinh^{-1} z = \log [z + \sqrt{z^2 + 1}]$

(ii) $\cosh^{-1} z = \log [z + \sqrt{(z^2 - 1)}]$

(iii) $\tanh^{-1} z = \frac{1}{2} \log \left( \frac{1+z}{1-z} \right)$

(i) Let $\sinh^{-1} z = u$, then $z = \sinh u = \frac{1}{2} (e^u - e^{-u})$

or

$2z = e^u - e^{-u} \quad \text{or} \quad e^{2u} - 2ze^u - 1 = 0$

This being a quadratic in $e^u$, we have

$e^u = \frac{2z + \sqrt{4z^2 + 4}}{2} = z + \sqrt{z^2 + 1}$

Thus, taking the positive sign only, we have

$e^u = z + \sqrt{z^2 + 1} \quad \text{or} \quad u = \log \left[ z + \sqrt{z^2 + 1} \right]$

Similarly, we can establish (ii).

(iii) Let $\tanh^{-1} z = u$, then $z = \tanh u$

i.e.,

$z = \frac{e^u - e^{-u}}{e^u + e^{-u}}$

Applying componendo and dividendo, we get

$\frac{1+z}{1-z} = e^{2u}$

or

$2u = \log \left( \frac{1+z}{1-z} \right) \quad \text{whence follows the result.}$

Logarithmic Function

Logarithmic Function:

1. If $z = (x + iy)$ and $w = (u + iv)$ be so related that $e^w = z$, then $w$ is said to be a logarithm of $z$ to the base $e$ and is written as $w = \log_e z\; \; \; text{i}$. Also

   $e^{w + 2in\pi} = e^w \cdot e^{2in\pi} = z \quad \left[ \because e^{2in\pi} = 1 \right]$ Thus,

   $\log z = w + 2in\pi \; \; \; \; \; \text{ii}$

   i.e., the logarithm of a complex number has an infinite number of values and is, therefore, a multi-valued function.

   The general value of the logarithm of $z$ is written as $\text{Log } z$ (beginning with capital $L$) so as to distinguish it from its principal value which is written as $\log z$. This principal value is obtained by taking $n = 0$ in $\text{Log } z$.

   Thus from (i) and (ii),

   $\text{Log } (x + iy) = 2in\pi + \log (x + iy)$

2. Real and imaginary parts of Log $(x + iy)$.

   $\text{Log } (x + iy) = 2in\pi + \log (x + iy)$

   $= 2in\pi + \log \left[ r (\cos \theta + i \sin \theta) \right]$

   $= 2in\pi + \log \left( r e^{i\theta} \right)$

   $= 2in\pi + \log r + i\theta = \log \sqrt{x^2 + y^2} + i \left[ 2n\pi + \tan^{-1} \left( \frac{y}{x} \right) \right]$

   $\text{Put } x = r \cos \theta, \, y = r \sin \theta \text{ so that } r = \sqrt{x^2 + y^2} \text{ and } \theta = \tan^{-1} \left( \frac{y}{x} \right)$

3. Real and imaginary parts of $(\alpha + i \beta)^{\gamma + i \delta}$.

   $(\alpha + i \beta)^{\gamma + i \delta} = e^{(\gamma + i \delta) \log (\alpha + i \beta)} = e^{(\gamma + i \delta) [\log r + i(\theta + 2n\pi)]}$

   $= e^{(\gamma + i \delta) [\log r + i\theta]} = e^{[\gamma \log r - \delta \theta]} e^{i[\delta \log r + \gamma \theta + 2n\pi \gamma]}$

   $= e^A [\cos B + i \sin B]$

   where $A = x \log r - y(2n\pi + \theta)$ and $B = y \log r + x(2n\pi + \theta)$.

   $\therefore$ the required real part = $e^A \cos B$ and the imaginary part = $e^A \sin B$.

Analytic Functions

Analytic Functions:

A function $f(z)$ which is single-valued and possesses a unique derivative with respect to $z$ at all points of a region $R$, is called an analytic function of $z$ in that region. An analytic function is also called a regular function or a holomorphic function.

A function which is analytic everywhere in the complex plane, is known as an entire function. As derivative of a polynomial exists at every point, a polynomial of any degree is an entire function.

A point at which an analytic function ceases to possess a derivative is called a singular point of the function.

Thus if $u$ and $v$ are real single-valued functions of $x$ and $y$ such that $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, $\partial v/\partial y$ are continuous throughout a region $R$, then the Cauchy-Riemann equations

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} \; \; \; \; \text{(1)}$

are both necessary and sufficient conditions for the function $f(z) = u + iv$ to be analytic in $R$.

The real and imaginary parts of an analytic function are called conjugate functions. The relation between two conjugate functions is given by the C-R equation (1).

Cauchy - Riemann Equations