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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.
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.
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:
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:
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.
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.
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.
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.
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.
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.
The sumA+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
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:
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]
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: