Applied Mathematics/Printable version

Applied Mathematics

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The Basics of Theory of The Fourier Transform

The Basics edit

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

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   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

  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



Integral calculus edit

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

Definite integral is defined as follows:

Examples edit



Practice problems edit


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  .

Fourier Series

For the function  , Taylor expansion is possible.


This is the Taylor expansion of  . On the other hand, more generally speaking,   can be expanded by also Orthogonal f

Fourier series edit

For the function   which has   for its period, the series below is defined:


This series is referred to as Fourier series of  .   and   are called Fourier coefficients.


where   is natural number. Especially when the Fourier series is equal to the  , (1) is called Fourier series expansion of  . Thus Fourier series expansion is defined as follows:


General Fourier Transform

Fourier Transform edit

The Fourier transform relates the function's time domain, shown in red, to the function's frequency domain, shown in blue. The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain.

Fourier Transform is to transform the function which has certain kinds of variables, such as time or spatial coordinate,   for example, to the function which has variable of frequency.

Definition edit


This integral above is referred to as Fourier integral, while   is called Fourier transform of  .   denotes "time".   denotes "frequency".

On the other hand, Inverse Fourier transform is defined as follows:


In the textbooks of universities, the Fourier transform is usually introduced with the variable Angular frequency  . In other word,   is substituted to (1) and (2) in the books. In that case, the Fourier transform is written in two different ways.





Fourier Sine Series

Fourier sine series edit

The series below is called Fourier sine series.




Fourier Cosine Series

Fourier cosine series edit

The series below is called Fourier cosine series.




Fourier Integral Transforms

Let   and



Then we have the functions below.


This function   is referred to as Fourier integral.


This function   is referred to as fourier transform as we previously learned.

Parseval's Theorem

Parseval's theorem edit


where   represents the continuous Fourier transform of x(t) and f represents the frequency component of x. The function above is called Parseval's theorem.

Derivation edit

Let   be the complex conjugation of  .


Here, we know that   is equal to the expansion coefficient of   in fourier transforming of  .
Hence, the integral of   is




Bessel Functions

Bessel Functions edit

The equation below is called Bessel's differential equation.


The two distinctive solutions of Bessel's differential equation are either one of the two pairs: (1)Linear combination of Bessel function(also known as Bessel function of the first kind) and Neumann function(also known as Bessel function of the second kind) (2)Linear combination of Hankel function of the first kind and Hankel function of the second kind. Bessel function (of the first kind) is denoted as  . Bessel function is defined as follow:




Γ(z) is the gamma function.   is the imaginary unit.   is the Neumann function(or Bessel function of the second kind).   is the Hankel functions. If n is an integer, the Bessel function of the first kind is an entire function.

Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering.

Laplace Transforms involve a technique to change an expression into another form that is easier to work with using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, Laplace Transforms require only improper integration techniques to use. So you may run across them in first year calculus.

Notation: The Laplace Transform is denoted as  .

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

Definition edit

For a function  , using Napier's constant   and a complex number  , the Laplace transform   is defined as follows:


The parameter   is a complex number.

  with real numbers   and  .

This   is the Laplace transform of  .

Explanation edit

Here is what is going on.

Examples of Laplace transform edit

Examples of Laplace transform

In the above table,

  1.   and   are constants
  2.   is a natural number
  3.   is the Delta function
  4.   is the Heaviside function

ID Function Time domain
Laplace domain
Region of convergence
for causal systems
1 Ideal delay    
1a Unit impulse      
2 Delayed nth power with frequency shift      
2a nth Power      
2a.1 qth Power      
2a.2 Unit step      
2b Delayed unit step      
2c Ramp      
2d nth Power with frequency shift      
2d.1 Exponential decay      
3 Exponential approach      
4 Sine      
5 Cosine      
6 Hyperbolic sine      
7 Hyperbolic cosine      
8 Exponentially-decaying sine      
9 Exponentially-decaying cosine      
10 nth Root      
11 Natural logarithm      
12 Bessel function
of the first kind, of order n
13 Modified Bessel function
of the first kind, of order n
14 Bessel function
of the second kind, of order 0
15 Modified Bessel function
of the second kind, of order 0
16 Error function      
17 Constant    
Explanatory notes:

  •   represents the Heaviside step function.
  •   represents the Dirac delta function.
  •   represents the Gamma function.
  •   is the Euler-Mascheroni constant.

  •  , a real number, typically represents time,
    although it can represent any independent dimension.
  •   is the complex angular frequency.
  •  ,  ,  ,   and   are real numbers.
  •  is an integer.
  • A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.

Examples edit

1. Calculate   (where   is a constant) using the integral definition.



2. Calculate   using the integral definition.



Complex Integration

Complex integration edit

On the piecewise smooth curve    , suppose the function f(z) is continuous. Then we obtain the equation below.


where   is the complex function, and   is the complex variable.

Proof edit





The right side of the equation is the real integral, therefore, according to calculus, the relationship below can be applied.




This completes the proof.

The Basics

The Basics of linear algebra edit


A matrix is composed of a rectangular array of numbers arranged in rows and columns. The horizontal lines are called rows and the vertical lines are called columns. The individual items in a matrix are called elements. The element in the i-th row and the j-th column of a matrix is referred to as the i,j, (i,j), or (i,j)th element of the matrix. To specify the size of a matrix, a matrix with m rows and n columns is called an m-by-n matrix, and m and n are called its dimensions.

Basic operation[1] edit

Operation Definition Example
Addition The sum A+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)i,j = Ai,j + Bi,j, where 1 ≤ im and 1 ≤ jn.


Scalar multiplication The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:
(cA)i,j = c · Ai,j.
Transpose The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
(AT)i,j = Aj,i.

Practice problems edit


Matrix multiplication edit

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B[2]


Schematic depiction of the matrix product AB of two matrices A and B.

Example edit




Practice Problems edit



Dot product edit

A row vector is a 1 × m matrix, while a column vector is a m × 1 matrix.

Suppose A is row vector and B is column vector, then the dot product is defined as follows;




Suppose   and   The dot product is


Example edit

Suppose   and  




Practice problems edit

(1)   and  


(2)   and  


Cross product edit

Cross product is defined as follows:


Or, using detriment,


where   is unit vector.

References edit

  1. Sourced from Matrix (mathematics), Wikipedia, 28th March 2013.
  2. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.
  3. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.

Lagrange Equations

Lagrange Equation edit


The equation above is called Lagrange Equation.

Let the kinetic energy of the point mass be   and the potential energy be  .
  is called Lagrangian. Then the kinetic energy is expressed by




Hence the Lagrangian   is


Therefore   relies on only   and  .   relies on only   and  . Thus


In the same way, we have


The State Equation

Ideal gas law edit

As an experimental rule, about gas, the equation below is found.



  = Pressure (absolute)
  = Volume
  = Number of moles of a substance
  = Absolute temperature
  = Gas constant

The gas which strictly follows the law (1) is called ideal gas. (1) is called ideal gas law. Ideal gas law is the state equation of gas. In high school, the ideal gas constant below might be used:


But, in university, the gas constant below is often used: