
Circular/ Trigonometric Functions
The sine and cosine functions are known as circular functions, because represents a point on the unit circle. Also,
From Euler’s formula, ,
These definitions are valid for any number .

Hyperbolic Functions
The hyperbolic functions parameterize the hyperbola (just like circular functions parameterize the circle). They appear in solutions of many linear differential equations. They are defined as:

Relationship between Circular and Hyperbolic Functions

Other Formulas
By using the definitions, we can also write down the expressions for derivatives and integrations of hyperbolic functions.

Inverse Hyperbolic Functions

Important Trigonometric Formulas
Also see these: post 1 and post 2.

Logarithms of Complex Numbers
We use the polar representation of complex numbers. Let
Thus, of a complex number is multivalued. The principal value of log is

Types of Problems on Logarithms of Complex Numbers
The problems ask for separating real and imaginary parts of a complicated expression.
I) : Use
II) : Use change of base rule.
III) : Take on both sides.