Applied Mathematics/The Basics of Theory of The Fourier Transform

The Basics

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

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

Differentiation is the process of finding a derivative of the function $f(x)$ in the independent input x. The differentiation of $f(x)$ is denoted as $f'(x)$ or ${\frac {d}{dx}}f(x)$. Both of the two notations are same meaning.

Differentiation is manipulated as follows:
${\frac {d}{dx}}(x^{3}+1)$
$=3x^{3-1}$
$=3x^{2}$

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 $x$ is just removed in differentiation.

Examples

${\frac {d}{dx}}(x^{5}+x^{2}+28)$
$=5x^{5-1}+2x^{2-1}$ 28 doesn't have the variable x, so 28 is removed
$=5x^{4}+2x^{1}$
$=5x^{4}+2x$

${\frac {d}{dx}}(x^{7}+x^{4}+x+7)$
$=7x^{7-1}+4x^{4-1}+1x^{1-1}$ 7 doesn't have the variable x, so 7 is removed
$=7x^{6}+4x^{3}++1x^{0}$
$=7x^{6}+4x^{3}+1$

Practice problems

(1)${\frac {d}{dx}}(x^{4}+15)=$

(2)${\frac {d}{dx}}(x^{5}+x^{3}+x)=$

Integral calculus

If you differentiate $f(x)=x^{3}$ or $f(x)=x^{3}-2$, each of them become $f'(x)=3x^{2}$. Then let's think of the opposite case. A function is provided, and when the function is differentiated, the function became $f'(x)=3x^{2}$. What's the original function? To find the original function, the integral calculus is used. Integration of $f(x)$ is denoted as $\int f(x)dx$.

Integration is manipulated as follows:
$\int 3x^{2}dx$
$={\frac {3x^{2+1}}{2+1}}+C$
$=x^{3}+C$
$C$ denotes Constant in the equation.

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

$\int x^{n}dx={\frac {x^{n+1}}{n+1}}+C$

Definite integral

Definite integral is defined as follows:
$\int _{a}^{b}f(x)dx$
$=[F(x)]_{a}^{b}$
$=F(b)-F(a)$
where $F(x)=\int f(x)dx$

Examples

(1)$\int 4xdx$
$={\frac {4x^{1+1}}{1+1}}+C$
$=2x^{2}+C$

(2)$\int _{1}^{2}(2x+1)dx$
$=[x^{2}+x]_{1}^{2}$
$=(4+2)-(1+1)$
$=4$

Practice problems

(1)$\int 6xdx=$
(2)$\int _{0}^{2}(3x^{2}+3)dx=$

Euler's number "e"

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

$\int e^{x}dx=e^{x}+C$

${\frac {d}{dx}}e^{x}=e^{x}$

By the way, in Mathematics, $exp(x)$ denotes $e^{x}$.