# Applied Mathematics/The Basics of Theory of The Fourier Transform

## The Basics[edit | edit source]

The two most important things in Theory of The Fourier Transform are "differential calculus" and "integral calculus". The readers are required to learn "differential calculus" and "integral calculus" before studying the Theory of The Fourier Transform. Hence, we will learn them on this page.

## Differential calculus[edit | edit source]

**Differentiation** is the process of finding a derivative of the function in the independent input *x*. The differentiation of is denoted as or . Both of the two notations are same meaning.

Differentiation is manipulated as follows:

As you see, in differentiation, the number of the degree of the variable is multiplied to the variable, while the degree is subtracted one from itself at the same time. The term which doesn't have the variable is just removed in differentiation.

### Examples[edit | edit source]

28 doesn't have the variable *x*, so 28 is removed

7 doesn't have the variable *x*, so 7 is removed

### Practice problems[edit | edit source]

(1)

(2)

## Integral calculus[edit | edit source]

If you differentiate or , each of them become . Then let's think of the opposite case. A function is provided, and when the function is differentiated, the function became . What's the original function? To find the original function, the **integral calculus** is used. Integration of is denoted as .

Integration is manipulated as follows:

denotes Constant in the equation.

More generally speaking, the integration of f(x) is defined as:

### Definite integral[edit | edit source]

Definite integral is defined as follows:

where

### Examples[edit | edit source]

(1)

(2)

### Practice problems[edit | edit source]

(1)

(2)

## Euler's number "e"[edit | edit source]

Euler's number (also known as Napier's constant) has special features in differentiation and integration:

By the way, in Mathematics, denotes .