Complex Variables (original) (raw)

Last Updated : 10 Nov, 2025

In mathematics, complex analysis is known as the theory of functions. Complex variables form the foundation of a critical branch of mathematics known as complex analysis, which extends the principles of calculus to functions of a complex variable. Complex variables are variables that can take on complex numbers as values.

A complex number is a number that can be expressed in the form z = x + iy, where x and y are real numbers, and i is the imaginary unit, which satisfies i2 = −1. A complex variable z is typically expressed in the form:

**z = x + iy

Where,

• x is the real part of the complex variable.
• y is the imaginary part of the complex variable.
• i is the imaginary unit, which satisfies the equation i2 = −1.

Representation of Complex Variables

Some of the common representations of complex variables are:

• Cartesian Form
• Polar Form
• Exponential Form

Cartesian (Rectangular) Form

In complex numbers, the Cartesian form (also called rectangular form) represents a complex number using its real and imaginary components. In this form, a complex variable z is expressed as:

z = x + iy

Where,

• x = real part of z.
• y = imaginary part of z.
• i = imaginary unit, satisfying i2 = −1.

Polar Form

A complex number can also be represented using its magnitude and angle, instead of its real and imaginary parts. This representation is called the polar form. The relationship is given by:

z = r(cos⁡θ + isin⁡θ)

Where:

• r = ∣z∣ = √[x2 + y2 ] is the modulus of z.
• θ = arg⁡(z) = tan⁡−1(y/x) is the argument of z, representing the angle with the positive real axis.

Using Euler's formula, this can also be written as:

z = reiθ

Exponential Form

A compact and elegant representation using Euler's formula is the exponential form:

**z = re iθ

Where r and θ are as defined above.

Operations on Complex Variables

Similar to any other variable, complex variables undergo the same operations, i.e.,

• Addition
• Subtraction
• Multiplication
• Division
• Conjugate

Let's discuss these operations for complex variables with examples.

Addition and Subtraction

To add or subtract two complex numbers, we simply add or subtract their real and imaginary parts separately. If we have two complex numbers z1 = a + ib and z2 = c + id, their sum is given by:

**z 1 ± z 2 **= (a ± c) + i(b ± d)

Let's consider some examples for better understanding.

**Example: If z 1 **= 3 + 4i and z 2 **= 1 + 2i, then calculate

• **z 1 + z 2
• **z 1 + z 2

**Solution:

• z1 + z2 = (3 + 1) + i(4 + 2) = 4 + 6i
• z1 - z2 = (3 - 1) + i(4 - 2) = 2 + 2i

Multiplication and Division

**To multiply two complex numbers, we use the distributive property and the fact that i2 = −1. For z1 = a + ib and z2 = c + id:

z1 ⋅ z2 = (a + ib)(c + id) = ac + iad + ibc + bd i2

Since i2 = −1, this simplifies to:

z1 ⋅ z2 = ac - bd + iad + ibc

⇒ z1 ⋅ z2 = (ac − bd) + (ad + bc)i

**To divide one complex number by another, we multiply the numerator and the denominator by the conjugate of the denominator. For z1 = a + ib and z2 = c + id:

\frac{z_1}{z_2} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}

The denominator simplifies to c2 + d2, and the numerator is computed as follows:

(a + bi)(c − id) = ac − iad + ibc − bdi2 = ac + bd + i(bc − ad)(a + bi)

So,

\frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2}

Let's consider an example for both

**Example: If z1 = 3 + i4 and z2 = 1 + i2, then find z1 ⋅ z2.

**Solution:

z1 ⋅ z2 = (3 ⋅ 1 − 4 ⋅ 2) + i(3 ⋅ 2 + 4 ⋅ 1) = (3 − 8) + (6 + 4)i = −5 + 10i

**Example: If z1 = 3 + 4i and z2 = 1 + 2i, then find z1/z2.

**Solution:

\frac{z_1}{z_2} = \frac{(3 + 4i)(1 - 2i)}{(1 + 2i)(1 - 2i)} = \frac{(3 \cdot 1 + 4 \cdot 2) + (4 \cdot 1 - 3 \cdot 2)i}{1^2 + 2^2} = \frac{(3 + 8) + (4 - 6)i}{1 + 4} = \frac{11 - 2i}{5} = \frac{11}{5} - \frac{2}{5}i = 2.2 - 0.4i

Conjugate of a Complex Variable

The conjugate of a complex number z = a + ib is denoted by \overline{z} and is defined as:

\overline{z} = a - ib

**Example: If z = 3 + i4, then find its conjugate.

**Solution:

**Given: z = 3 + i4

Thus, \overline{z} = 3 - i4

Functions of Complex Variables

There are various types of functions that involve complex variables. Some of the functions with complex variables are discussed below:

Analytic Functions

A function f(z) is said to be analytic (or holomorphic) at a point z0 ​if it is differentiable at z0​ and in a neighborhood around z0​. If a function is analytic at every point in its domain, it is called an entire function.

Elementary Functions

Some of the common elementary functions are:

**Related Articles:

• **Complex Numbers
• **Imaginary Numbers
• **Polar and Exponential Forms of Complex Numbers
• **Euler's Formula