Applied Mathematics/The Basics of Theory of The Fourier Transform

The BasicsEdit

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 calculusEdit

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.


ExamplesEdit

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

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

Practice problemsEdit

(1) 

(2) 

Integral calculusEdit

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 integralEdit

Definite integral is defined as follows:
 
 
 
where  

ExamplesEdit

(1) 
 
 

(2) 
 
 
 

Practice problemsEdit

(1) 
(2) 

Euler's number "e"Edit

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

 
 

By the way, in Mathematics,   denotes  .