Applied Mathematics/Printable version


Applied Mathematics

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The Basics of Theory of The Fourier Transform

The Basics

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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

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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

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

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

Practice problems

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(1) 

(2) 

Integral calculus

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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

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Definite integral is defined as follows:
 
 
 
where  

Examples

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(1) 
 
 

(2) 
 
 
 

Practice problems

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(1) 
(2) 

Euler's number "e"

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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

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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

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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

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 ...(1)

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:

  ...(2)

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.

1.

 
 

2.

 
 


Fourier Sine Series

Fourier sine series

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The series below is called Fourier sine series.

 

where

 


Fourier Cosine Series

Fourier cosine series

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The series below is called Fourier cosine series.

 

where

 


Fourier Integral Transforms

Let   and

suppose

 .

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

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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

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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

 
 
 
 
 

Hence

 


Bessel Functions

Bessel Functions

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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:

 
 

where

 
 
 

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

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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

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Here is what is going on.

Examples of Laplace transform

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Examples of Laplace transform
   
   
   
   
   
   
   
   
   
    (n>0)
   
   

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

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

 

 


2. Calculate   using the integral definition.

 

 


Complex Integration

Complex integration

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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

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Let

 
 

Then

 
 
 

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

 

Hence

 
 
 
 
 

This completes the proof.


The Basics

The Basics of linear algebra

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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]

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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

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(1)  
(2)  
(3)  

Matrix multiplication

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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]

 [3]


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



Example

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Practice Problems

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(1)  

(2)  


Dot product

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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;

 

or

 


Suppose   and   The dot product is

 

Example

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Suppose   and  

 

 
 

 

Practice problems

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(1)   and  

 

(2)   and  

 

Cross product

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Cross product is defined as follows:

 

Or, using detriment,

 

where   is unit vector.

References

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  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

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where  
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

 
 

Thus

 
 

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

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

 ...(1)

where

  = 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: