Complex Numbers and Complex Functions

1. Complex Numbers

A complex number is a number that can be expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying the equation \( i^2 = -1 \).

1.1. Definitions and Basic Properties

The complex number \( z = a + bi \) consists of two parts: the real part, denoted as \( \text{Re}(z) = a \), and the imaginary part, denoted as \( \text{Im}(z) = b \).

Two complex numbers \( z_1 = a_1 + b_1i \) and \( z_2 = a_2 + b_2i \) are equal if and only if their real parts are equal and their imaginary parts are equal, i.e., \( a_1 = a_2 \) and \( b_1 = b_2 \).

1.2. Operations on Complex Numbers

Addition: \( (a_1 + b_1i) + (a_2 + b_2i) = (a_1 + a_2) + (b_1 + b_2)i \).
Example: \( (2 + 3i) + (4 - i) = (2+4) + (3-1)i = 6 + 2i \).

Subtraction: \( (a_1 + b_1i) - (a_2 + b_2i) = (a_1 - a_2) + (b_1 - b_2)i \).
Example: \( (5 + 2i) - (1 + 3i) = (5-1) + (2-3)i = 4 - i \).

Multiplication: \( (a_1 + b_1i)(a_2 + b_2i) = a_1a_2 + a_1b_2i + b_1ia_2 + b_1ib_2i = (a_1a_2 - b_1b_2) + (a_1b_2 + b_1a_2)i \).
Example: \( (2 + 3i)(1 - i) = (2 \times 1 - 3 \times (-1)) + (2 \times (-1) + 3 \times 1)i = (2 + 3) + (-2 + 3)i = 5 + i \).

Division: To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \( z = a + bi \) is \( \bar{z} = a - bi \).
\( \frac{a_1 + b_1i}{a_2 + b_2i} = \frac{(a_1 + b_1i)(a_2 - b_2i)}{(a_2 + b_2i)(a_2 - b_2i)} = \frac{(a_1a_2 + b_1b_2) + (b_1a_2 - a_1b_2)i}{a_2^2 + b_2^2} = \frac{a_1a_2 + b_1b_2}{a_2^2 + b_2^2} + \frac{b_1a_2 - a_1b_2}{a_2^2 + b_2^2}i \).
Example: \( \frac{2 + i}{1 - i} = \frac{(2 + i)(1 + i)}{(1 - i)(1 + i)} = \frac{(2 \times 1 - 1 \times 1) + (2 \times 1 + 1 \times 1)i}{1^2 + (-1)^2} = \frac{(2-1) + (2+1)i}{1+1} = \frac{1 + 3i}{2} = \frac{1}{2} + \frac{3}{2}i \).

1.3. Geometric Representation (Argand Plane)

A complex number \( z = a + bi \) can be represented as a point \( (a, b) \) in a two-dimensional Cartesian coordinate system called the Argand plane or complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

The distance from the origin \( (0, 0) \) to the point \( (a, b) \) is called the modulus or magnitude of the complex number, denoted by \( |z| \).
\( |z| = \sqrt{a^2 + b^2} \).

The angle \( \theta \) that the line segment from the origin to \( (a, b) \) makes with the positive real axis is called the argument or phase of the complex number, denoted by \( \arg(z) \).
\( \theta = \arctan\left(\frac{b}{a}\right) \), taking into account the quadrant of \( (a, b) \). The principal argument is usually taken in the interval \( (-\pi, \pi] \).

1.4. Polar and Exponential Forms

A complex number \( z = a + bi \) can be expressed in polar form as:
\( z = r(\cos \theta + i \sin \theta) \), where \( r = |z| \) and \( \theta = \arg(z) \).

Using Euler's formula, \( e^{i\theta} = \cos \theta + i \sin \theta \), the exponential form is:
\( z = re^{i\theta} \).

De Moivre's Theorem: For any complex number \( z = r(\cos \theta + i \sin \theta) \) and any integer \( n \),
\( z^n = r^n(\cos(n\theta) + i \sin(n\theta)) \). In exponential form: \( (re^{i\theta})^n = r^n e^{in\theta} \).
This theorem is crucial for calculating powers and roots of complex numbers.

Roots of Complex Numbers: To find the \( n \)-th roots of a complex number \( z = re^{i\theta} \), we solve \( w^n = z \). The \( n \) distinct roots are given by:
\( w_k = r^{1/n} e^{i(\frac{\theta + 2k\pi}{n})} \) for \( k = 0, 1, 2, \dots, n-1 \).

1.5. Powers of i

The powers of the imaginary unit \( i \) follow a cyclic pattern:
\( i^1 = i \)
\( i^2 = -1 \)
\( i^3 = i^2 \times i = -i \)
\( i^4 = i^2 \times i^2 = (-1)(-1) = 1 \)
\( i^5 = i^4 \times i = 1 \times i = i \)
The cycle repeats every four powers. To find \( i^n \), divide \( n \) by 4 and look at the remainder:

• If remainder is 0, \( i^n = 1 \)
• If remainder is 1, \( i^n = i \)
• If remainder is 2, \( i^n = -1 \)
• If remainder is 3, \( i^n = -i \)

2. Complex Functions

A complex function is a function whose domain and/or range are subsets of the complex numbers. A complex function \( f \) typically maps a complex number \( z \) to another complex number \( w \). We write this as \( w = f(z) \).

2.1. Definition and Representation

If \( z = x + iy \) and \( w = u + iv \), where \( x, y, u, v \) are real numbers, then a complex function \( f(z) \) can be expressed in terms of its real and imaginary parts:
\( f(z) = f(x + iy) = u(x, y) + iv(x, y) \). Here, \( u(x, y) \) is the real part of \( f(z) \) and \( v(x, y) \) is the imaginary part of \( f(z) \).

Example: Consider the function \( f(z) = z^2 \).
Let \( z = x + iy \). Then \( f(z) = (x + iy)^2 = x^2 + 2xyi + (iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + i(2xy) \).
So, the real part is \( u(x, y) = x^2 - y^2 \) and the imaginary part is \( v(x, y) = 2xy \).

2.2. Types of Complex Functions

Polynomial Functions: \( P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0 \), where \( a_k \) are complex constants.
Example: \( f(z) = 3z^3 - 2z + 5 \).

Rational Functions: \( R(z) = \frac{P(z)}{Q(z)} \), where \( P(z) \) and \( Q(z) \) are polynomial functions, and \( Q(z) \neq 0 \).
Example: \( f(z) = \frac{z^2 + 1}{z - i} \). This function is defined for all \( z \neq i \).

Exponential Function: \( f(z) = e^z \).
If \( z = x + iy \), then \( e^z = e^{x+iy} = e^x e^{iy} = e^x (\cos y + i \sin y) \).
The real part is \( u(x, y) = e^x \cos y \) and the imaginary part is \( v(x, y) = e^x \sin y \).

Trigonometric Functions: Defined using the exponential function:
\( \cos z = \frac{e^{iz} + e^{-iz}}{2} \)
\( \sin z = \frac{e^{iz} - e^{-iz}}{2i} \)
\( \tan z = \frac{\sin z}{\cos z} \), etc. Note that these functions can take values outside the range [-1, 1] for complex arguments. For example, \( \cos(i) = \cosh(1) = \frac{e + e^{-1}}{2} \approx 1.54 \).

Logarithmic Function: The complex logarithm is the inverse of the exponential function. For \( z = re^{i\theta} \), the principal value of the logarithm is:
\( \text{Log}(z) = \ln |z| + i \arg(z) = \ln r + i\theta \).
The general logarithm includes all possible values of the argument:
\( \log(z) = \ln |z| + i (\arg(z) + 2k\pi) = \ln r + i(\theta + 2k\pi) \), where \( k \) is an integer.
Example: \( \log(1+i) \). Here \( r = |1+i| = \sqrt{1^2 + 1^2} = \sqrt{2} \) and \( \theta = \arg(1+i) = \frac{\pi}{4} \).
The principal value is \( \text{Log}(1+i) = \ln \sqrt{2} + i \frac{\pi}{4} = \frac{1}{2}\ln 2 + i \frac{\pi}{4} \).
The general values are \( \log(1+i) = \frac{1}{2}\ln 2 + i (\frac{\pi}{4} + 2k\pi) \).

Power Function: \( f(z) = z^c \), where \( c \) is a complex number.
\( z^c = e^{c \log z} \). Since \( \log z \) is multi-valued, \( z^c \) is generally multi-valued. The principal value is obtained using the principal value of the logarithm: \( z^c = e^{c \text{Log } z} \).
Example: \( i^i \). Using the principal value:
\( i^i = e^{i \text{Log}(i)} \). Here \( |i| = 1 \) and \( \arg(i) = \frac{\pi}{2} \).
\( \text{Log}(i) = \ln(1) + i \frac{\pi}{2} = 0 + i \frac{\pi}{2} = i \frac{\pi}{2} \).
So, \( i^i = e^{i (i \frac{\pi}{2})} = e^{-\frac{\pi}{2}} \). This is a real number!

2.3. Analyticity and Cauchy-Riemann Equations

A complex function \( f(z) = u(x, y) + iv(x, y) \) is said to be differentiable at a point \( z_0 \) if the limit
\( f'(z_0) = \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z} \) exists.

A function \( f(z) \) is analytic (or holomorphic) in an open region \( D \) if it is differentiable at every point in \( D \).

The Cauchy-Riemann equations provide a necessary condition for a function to be analytic. If \( f(z) = u(x, y) + iv(x, y) \) is differentiable at \( z = x + iy \), then the partial derivatives of \( u \) and \( v \) must satisfy:
\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and
\( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

If the first partial derivatives of \( u \) and \( v \) are continuous in a region \( D \), then the Cauchy-Riemann equations are also a sufficient condition for analyticity.

2.4. Harmonic Functions

If a function \( f(z) = u(x, y) + iv(x, y) \) is analytic in a region, then both its real part \( u(x, y) \) and its imaginary part \( v(x, y) \) are harmonic functions in that region.

A function \( \phi(x, y) \) is harmonic if it satisfies Laplace's equation:
\( \nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 \).

If \( u(x, y) \) is harmonic, its harmonic conjugate \( v(x, y) \) can be found using the Cauchy-Riemann equations. The function \( f(z) = u + iv \) is then analytic.

Finding Harmonic Conjugates: Given a harmonic function \( u(x, y) \), we want to find \( v(x, y) \) such that \( u + iv \) is analytic. We know \( v_y = u_x \) and \( v_x = -u_y \). 1. Integrate \( u_x \) with respect to \( y \) to get an expression for \( v \) in terms of \( y \) and \( x \): \( v(x, y) = \int u_x(x, y) \, dy + g(x) \). 2. Differentiate this expression for \( v \) with respect to \( x \): \( v_x = \frac{\partial}{\partial x} \left( \int u_x(x, y) \, dy \right) + g'(x) \). 3. Set this equal to \( -u_y(x, y) \) and solve for \( g'(x) \). 4. Integrate \( g'(x) \) to find \( g(x) \). 5. Substitute \( g(x) \) back into the expression for \( v(x, y) \).

2.5. Conformal Mapping

An analytic function \( f(z) \) defines a transformation (or mapping) from the \( z \)-plane to the \( w \)-plane. If \( f'(z_0) \neq 0 \), the mapping \( w = f(z) \) is conformal at \( z_0 \).

Conformal mapping preserves angles between curves, both in magnitude and orientation. This property is extremely useful in solving problems involving potential theory (like heat distribution or fluid flow) in complex domains.

If \( f'(z_0) = 0 \), the mapping is not conformal at \( z_0 \). The angle is magnified by the order of the zero of \( f'(z) \) at \( z_0 \).

2.6. Complex Integration

Integration of complex functions is performed along curves (paths) in the complex plane. A path \( \gamma \) can be parameterized as \( z(t) = x(t) + iy(t) \) for \( a \le t \le b \).

The complex line integral of a function \( f(z) \) along a path \( \gamma \) is defined as:
\( \int_{\gamma} f(z) \, dz = \int_{a}^{b} f(z(t)) z'(t) \, dt \).

Cauchy's Integral Theorem: If \( f(z) \) is analytic in a simply connected domain \( D \), then for any simple closed curve \( \gamma \) lying entirely within \( D \),
\( \oint_{\gamma} f(z) \, dz = 0 \).

Cauchy's Integral Formula: If \( f(z) \) is analytic in a simply connected domain \( D \), and \( \gamma \) is a simple closed curve in \( D \), and \( z_0 \) is any point inside \( \gamma \), then:
\( f(z_0) = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{z - z_0} \, dz \).
This formula allows us to calculate the value of an analytic function at a point using an integral around the point. It also leads to formulas for derivatives:
\( f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_{\gamma} \frac{f(z)}{(z - z_0)^{n+1}} \, dz \).