Complex Numbers

The Complex Plane

Complex Numbers

The field of complex numbers, denoted by , is the usual Euclidean space endowed with the additional operation of multiplication of vectors defined as follows: And we write for , for . Therefore denotes .

Alternative Algebraic Definition

Instead, algebraists define as (See here). This is equivalent to the above definition, but more algebraic.

Real Part & Imaginary Part

For any complex number , we call the real part of , denoted by , and the imaginary part of , denoted by . If , the complex number is called real. If , then is said to be imaginary.

Complex Conjugate

For complex number , we call the complex conjugate of .

Modulus

Define the modulus, or absolute value, of by

Proposition

yields a metric space by the metric defined by modulus of difference:

Proof Straightforward by definition.

Reciprocal and Ratio

The reciprocal or inverse of , denoted by , is the unique complex number such that . The ratio of two complex number is then naturally defined as

Proposition

.

Proof We have , hence .

The Argument

Proposition

Given a non-zero , it can be determined by its length and an angle .

Principle Argument

There are infinitely many angles that correspond to any arbitrary point , but we distinguish the appropriate angle in , called as the (principal) argument of . And denote with function . And define

Proposition

Function is not continuous.

Proof Consider the sequence and . Then , but and .

Proposition

Let be a continuous function with . Then for some independent of .

Exponential, Logarithm and Powers

Complex Exponential

Define the exponential function on complex numbers in the following way: where is the exponential of real numbers. Note that for any , we have

Remark

The differential equations uniquely determine the form of the function. We actually had no choice, as this function is the unique extension of to which is differentiable. See uniqueness theorem.

Proposition

For any , we have .

Proof We write and . Then using the fact that the above property holds for real exponential, and the angle sum formula for sine and cosine, we obtain that: $$\begin{aligned}e^{z_1}e^{z_2}& =e^{x_1}(\cos(y_1)+i\sin(y_1))e^{x_2}(\cos(y_2)+i\sin(y_2)) \&=e^{x_1+x_2}(\cos(y_1+y_2)+i\sin(y_1+y_2)) \&=e^{z_1+z_2}\end{aligned}$$$\square$

Proposition

When multiplying complex numbers, the lengths multiply and the angles add:

Proof Clearly obtained from the definition of complex multiplication and exponential.

Proposition

The complex exponential function is not injective.

Proof We have . Indeed if and only if and for some integer .

Complex Logarithm

Define the complex logarithm function as follows: where is the natural logarithm of a positive real number. And define

Proposition

Complex logarithm is the right inverse of complex exponential:

Proof

Proposition

The range of complex logarithm is the strip .

Proof The range of is , so the range of is .

Complex Powers

For any complex number and , we define .

Proposition

The following hold for complex powers:

• .
• If is a positive integer, then .

Proof

Trigonometric Functions

Complex Trigonometric Function

Define the complex trigonometric functions as follows:

Proposition

The complex trigonometric functions align with the real trigonometric functions when is purely real.

Proof Straightforward by definition.