§4 Fourier Transform
1. Fourier integral
[ Fourier integral ] A function that is absolutely integrable over any finite interval [- l , l ] , its Fourier series can be found ( §1 , 2 of this chapter )
_{} (1)
Assuming that the function is absolutely integrable over the infinite interval (- ) , in equation (1) , let l , obtain the Fourier integral of f ( x )_{}_{}_{}
_{}
[ Several forms of Fourier integral ]
Let the Fourier integral satisfy the convergence condition, then_{}
1 ^{o} = _{}_{}
2 ^{o} = ( _{}_{} outer integral is understood as integral in the sense of principal value )
3 ^{o} _{} is an even function:
_{}=_{}
4 ^{o} _{} is an odd function:
_{}=_{}
[ Convergence discriminant method of Fourier integral ] Assuming that the function is absolutely integrable, the imaginary value of the integral (1) is S _{0} . Suppose that the point x _{0} is continuous, or x _{0} is its first-type discontinuous point, and the continuous point is At point x _{0 }S _{0} = , while at the first type of discontinuity point x _{0} ,_{}_{}_{}_{}_{}_{}_{}_{}_{}_{}
S _{0} =_{}
1 ^{o} Dini discriminant decree , if for a , the integral _{}_{}
_{}
converges, then the Fourier integral converges at the point x _{0} and is equal to S _{0} ._{}_{}_{}
2 ^{o} Dirichlet - Rawdang discriminant If there is a bounded variation on an interval [ x _{0} - h , x _{0} + h ] with x _{0} as the midpoint, then its Fourier integral is at the point x _{0} converges and is equal to S _{0} . _{}_{}_{}_{}_{}_{}
3 ^{o} If the function has bounded variation on, while_{}_{}
_{}
Then the Fourier integral converges at any point x _{0 and is equal to }S _{0} ._{}_{}_{}
2. Fourier transform
The Fourier transform of [ Fourier transform and its inversion formula ] is_{}
_{}
The inversion formula of the Fourier transform is
_{}
The Fourier transform and inversion formula of [ Conditions for the existence of Fourier transform ] are meaningful _{}( only at the discontinuity point x _{0} of ) under the following two conditions, and the left end of the inversion formula should be equal to ):_{}_{}_{}
1 ^{o} _{} exists;
2 ^{o} _{} satisfies the Dirichlet condition: there are only a finite number of extreme points and only a finite number of discontinuities of the first kind ._{}_{}
[ Properties of Fourier Transform ] Let the Fourier transform of g ( x ) be F ( ) and G ( ) respectively, then_{}_{}_{}
1 ^{o} The Fourier transform of a linear a + b g ( x ) is a F ( ) + b G ( ) ( a , b are constants ) _{}_{}_{}
2 ^{o} The Fourier transform of convolution ( or convolution ) f ( x )*g( x )= _{} is
F ( ) G ( )_{}_{}_{}
3 ^{o} Parsepha equation _{}
4 ^{o} The Fourier transform of the flip f ( -x ) is F (- ). _{}
The ^{} Fourier transform of the 5o ^{conjugate} is . _{} _{}
^{} The Fourier transform of ^{6o} time - shift ( delay ) f ( x - x _{0} ) is . _{}_{}
7o ^{frequency} shift ( frequency modulation ) is the Fourier transform of ( is a constant ). _{}_{} _{}
[ Fourier transform table ]
_{} , _{}
_{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} _{} |
_{} _{} |
_{} _{} |
_{} |
_{} |
_{} _{} |
_{} _{} |
_{} _{} |
_{} _{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} |
_{} |
_{} _{} |
_{} |
_{} |
_{} |
_{} _{} |
_{} |
_{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
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_{} _{} |
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_{} _{} |
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_{} _{} |
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_{} _{} |
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_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} |
_{} ( g is Euler's constant ) |
_{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
3. Fourier cosine transform
[ Fourier cosine transform and its inversion formula ] The Fourier cosine transform of f ( x ) is
_{}
The inversion formula of the Fourier cosine transform is:
_{}
[ Existence condition of Fourier cosine transform ] is the same as the Fourier integral convergence condition .
[ Properties of Fourier Cosine Transform ]
1 ^{o} If it is the Fourier cosine transform of f ( x ) , then it is the Fourier cosine transform of ._{}_{}_{}
2 ^{o} If f ( x ) is an even function, then ._{}
^{} _{} The Fourier cosine transform of 3 ^{o} ( a > 0) is ._{}
[ Fourier cosine transform table ]
_{}, _{}
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} _{} |
_{} |
_{} |
_{} |
_{} |
_{} |
Four, Fourier sine transform
[ Fourier sine transform and its inversion formula ] The Fourier sine transform of f ( x ) is
_{}
The inversion formula of the Fourier sine transform is
_{}
[ Existence condition of Fourier sine transform ] is the same as the Fourier integral convergence condition .
[ Properties of Fourier Sine Transform ]
1 ^{o} If it is the Fourier sine transform of f ( x ) , then it is the Fourier sine transform of ._{}_{}_{}
2 ^{o} If f ( x ) is an odd function, then ._{}
^{} _{} The Fourier sine transform of 3 ^{o} ( a > 0) is ._{}
[ Fourier sine transformation table ]
_{}, _{}
_{} |
_{} |
_{} _{} |
_{} |
_{} |
_{} |
_{} _{} |
_{} |
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_{} |
_{} _{} |
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_{} _{} |
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_{} _{} |
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_{} _{} |
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_{} _{} |
_{} |
_{} |
_{} ( _{}for Euler's constant ) |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
5. Finite Fourier Cosine Transform
[ Finite Fourier cosine transform and its inversion formula ] Let f ( x ) satisfy the Dirichlet condition in the interval ( see this section, two ) , then the finite Fourier cosine transform of f ( x ) is_{}
_{}
The inversion formula of the finite Fourier cosine transform is:
at each successive point of f ( x ) in the interval_{}
_{}
At the discontinuity, the left-hand side of the equation is changed to ._{}
[ Finite Fourier Cosine Transform Table ]
_{}, _{}
_{} |
_{} |
1 |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} |
_{} _{} |
_{} _{}, _{} |
_{} ( m is an integer ) |
_{} |
6. Finite Fourier Sine Transform
[ Finite Fourier sine transform and its inversion formula ] Let f ( x ) satisfy the Dirichlet condition in the interval ( see this section, 2 ) , then the finite Fourier sine transform of f ( x ) is_{}
_{}
The inversion formula of the finite Fourier sine transform is:
at each successive point of f ( x ) on the interval_{}
_{}
At the discontinuity, the left-hand side of the equation is changed to ._{}
[ Finite Fourier Sine Transform Table ]
_{}, _{}
_{} |
_{} |
1 |
_{} |
_{} |
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_{} |
_{}, _{} |
_{} ( m is an integer ) |
_{} |
_{} ( m is an integer ) |
_{} |
Seven, double Fourier transform and its inversion formula
The double Fourier transform of f ( x , y ) is
_{}
The inversion formula of the double Fourier transform is:
_{}