General Mathematical Identities for Analytic Functions: Integral transforms

General Identities

General Mathematical Identities for Analytic Functions

General Identities

Integral transforms

Exponential Fourier transform

Definition

This formula is the definition of the exponential Fourier transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions for functions that do not grow faster than polynomials at .

Properties

Linearity

This formula reflects the linearity of the exponential Fourier transform.

Reflection

This formula is called the reflection property of the exponential Fourier transform.

Dilation

This formula reflects the scaling or dilation property of the exponential Fourier transform.

Shifting or translation

This formula reflects the shifting or translation property of the exponential Fourier transform.

Modulation

This formula reflects the modulation property of the exponential Fourier transform.

Power scaling

The power scaling property shows that multiplication of a function by corresponds to the derivative of the exponential Fourier transform.

This formula shows that multiplication of a function by corresponds to integration of the exponential Fourier transform.

Multiplication

The multiplication property shows that the exponential Fourier transform of a product gives the convolution of the exponential Fourier transform divided by .

Conjugation

This formula reflects the conjugation property of the exponential Fourier transform.

Derivative

The derivative property shows that the exponential Fourier transform of the derivative gives the product of the power function on the exponential Fourier transform.

Integral

This formula shows that the exponential Fourier transform of an integral gives the product of the power function and the exponential Fourier transform plus an expression that includes a Dirac delta function.

Parseval identity

This formula is called the Parseval identity.

This Bessel’s equality follows from the Parseval identity, when .

Convolution theorem

This Fourier convolution theorem or convolution (Faltung) theorem for the exponential Fourier transform shows that the Fourier transform of a convolution is equal to the product of the Fourier transform multiplied by .

Relations with other integral transforms

With inverse exponential Fourier transform

This formula reflects the relation between direct and inverse exponential Fourier transforms. In the point , where has a jump discontinuity the composition of the inverse and direct exponential Fourier transforms converges to the mean .

With Fourier cosine and sine transforms

This formula shows how the exponential Fourier transform can be represented through cosine and sine Fourier transforms from even and odd parts.

With Laplace transform

This formula shows how the exponential Fourier transform can be represented through the Laplace transform.

With Mellin transform

This formula shows how the exponential Fourier transform can be represented through the Mellin transform.

With Z-transform

This formula shows how the exponential Fourier transform can be represented through the Z‐transform.

Inverse exponential Fourier transform

Definition

This formula is the definition of the inverse exponential Fourier transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions.

Relations with other integral transforms

With exponential Fourier transform

This near‐equivalence identity shows that the inverse exponential Fourier transform in the point coincides with the direct Fourier transform in the point .

This formula reflects the relation between the direct and inverse exponential Fourier transforms. In the point , where has a jump discontinuity, the composition of inverse and direct exponential Fourier transforms converges to the mean .

Multiple exponential Fourier transform

Definition

This formula is the definition of the double exponential Fourier transform of the function with respect to the variables , . If the integral does not converge, the value of is defined in the sense of generalized functions.

This formula is the definition of the multiple exponential Fourier transform of the function with respect to the variables , ,…, over . If this integral does not converge, the value of is defined in the sense of generalized functions.

Inverse multiple exponential Fourier transform

Definition

This formula is the definition of the inverse double exponential Fourier transform of the function with respect to the variables , . If this integral does not converge, the value of is defined in the sense of generalized functions.

This formula is the definition of the inverse multiple exponential Fourier transform of the function with respect to the variables , ,…, over . If this integral does not converge, the value of is defined in the sense of generalized functions.

Relations with other integral transforms

With multiple exponential Fourier transform

This near‐equivalence identity shows that the inverse multiple exponential Fourier transform in the point coincides with the direct multiple exponential Fourier transform at the point .

Fourier transform (continuous -> discrete)

Definition

General properties

Linearity

Reflection

Dilation

Shifting or translation

Modulation

Multiplication

Conjugation

Derivative

Grouping

Summation

Parseval identity

Convolution theorem

Relations with other integral transforms

With inverse Fourier transform (continuous -> discrete)

Inverse Fourier transform (continuous -> discrete)

Definition

Relations with other integral transforms

With Fourier transform (continuous -> discrete)

Fourier transform (discrete -> continuous)

Definition

General properties

Linearity

Reflection

Dilation

Shifting or translation

Modulation

Power scaling

Multiplication

Conjugation

Sampling

Zero packing

Parseval identity

Convolution theorem

Relations with other integral transforms

With inverse Fourier transform (discrete -> continuous)

Inverse Fourier transform (discrete -> continuous)

Definition

Relations with other integral transforms

With Fourier transform (discrete -> continuous)

Fourier transform (discrete -> discrete)

Definition

General properties

Linearity

Reflection

Dilation

Shifting or translation

Modulation

Multiplication

Conjugation

Repeat

Zero packing

Summation

Parseval identity

Convolution theorem

Relations with other integral transforms

With inverse Fourier transform (discrete -> discrete)

Inverse Fourier transform (discrete -> discrete)

Definition

Relations with other integral transforms

With Fourier transform (discrete -> discrete)

Fourier cosine transform

Definition

This formula is the definition of the Fourier cosine transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions.

General properties

Linearity

This formula reflects the linearity of the Fourier cosine transform.

Scaling

This formula reflects the scaling property of the Fourier cosine transform.

Modulation

This formula reflects the modulation property of the Fourier cosine transform.

Parity

This formula shows how the Fourier cosine transform can be applied to the difference between the even part of and the odd part of .

This formula shows how the Fourier cosine transform can be applied to the sum of the even part of and the odd part of .

Power scaling

This formula shows that multiplication of a function by corresponds to the derivative of the Fourier cosine transform.

This formula shows that multiplication of a function by corresponds to the derivative of the Fourier sine transform.

Derivative

This formula shows that the Fourier cosine transform of an even-order derivative gives the product of the power function with the Fourier cosine transform plus some even polynomial.

This formula shows that the Fourier cosine transform of an odd-order derivative gives the product of a power function with the Fourier sine transform plus some even polynomial.

Convolution related

This formula shows that the Fourier cosine transform of a convolution gives the product of Fourier cosine transforms multiplied by .

Integral

This formula shows that the Fourier cosine transform of an indefinite integral with a variable lower limit gives the product of the Fourier sine transforms by .

Limit at infinity

This Riemann–Lebesgue theorem shows that the Fourier cosine transform converges to zero as tends to infinity for some classes of the function .

Relations with other integral transforms

With inverse Fourier cosine transform

This formula reflects the relation between the direct and the inverse Fourier cosine transforms. In the point , where has a jump discontinuity, the composition of the inverse and the direct Fourier cosine transforms converges to the mean .

With exponential Fourier transform

This formula reflects the relation between the direct and the inverse exponential Fourier transforms.

With Laplace transform

This formula represents the Fourier cosine transform through Laplace transforms.

Inverse Fourier cosine transform

Definition

This formula is the definition of the inverse Fourier cosine transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions.

Relations with other integral transforms

With Fourier cosine transform

This formula shows that the inverse Fourier cosine transform coincides with the direct Fourier cosine transform.

Fourier sine transform

Definition

This formula is the definition of the Fourier sine transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions.

General properties

Linearity

This formula reflects the linearity of the Fourier sine transform.

Scaling

This formula reflects the scaling property of the Fourier sine transform.

Modulation

This formula reflects the modulation property of the Fourier sine transform.

Parity

This formula shows how the Fourier sine transform can be applied to the difference between the even part of and the odd part of .

This formula shows how the Fourier sine transform can be applied to a sum of the even part of and the odd part of .

Power scaling

This formula shows that multiplication of a function by corresponds to the derivative of the Fourier sine transform.

This formula shows that multiplication of a function by corresponds to the derivative of the Fourier cosine transform.

Derivative

This formula shows that the Fourier sine transform of an even-order derivative gives the product of the power function and the Fourier sine transform plus some odd‐order polynomial.

This formula shows that the Fourier sine transform of an odd-order derivative gives the product of the power function and the Fourier cosine transform plus some odd‐order polynomial.

Convolution related

This formula shows that the Fourier sine transform of a convolution gives the product of the Fourier sine and the Fourier cosine transforms multiplied by .

Integral

This formula shows that the Fourier sine transform of an indefinite integral with a variable lower limit gives the difference of the Fourier cosine transforms in and multiplied by .

Limit at infinity

The Riemann–Lebesgue theorem shows that the Fourier sine transform converges to zero as tends to infinity for some classes of function .

Relations with other integral transforms

With inverse Fourier sine transform

This formula reflects the relation between the direct and the inverse Fourier sine transforms. At the point , where has a jump discontinuity, the composition of the inverse and the direct Fourier sine transforms converges to the mean .

With exponential Fourier transform

This formula represents the Fourier sine transform through the exponential Fourier transform.

With Laplace transform

This formula represents the Fourier sine transform through the Laplace transforms.

Inverse Fourier sine transform

Definition

This formula is the definition of the inverse Fourier sine transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions.

Relations with other integral transforms

With Fourier sine transform

This formula shows that the inverse Fourier sine transform coincides with the direct Fourier sine transform.

Laplace transform

Definition

This formula is the definition of the Laplace transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions.

General properties

Linearity

This formula reflects the linearity of the Laplace transform.

Shift

This shift theorem shows that the Laplace transform of a product with an exponential function gives the Laplace transform in the shifted point.

Power scaling

This formula shows that differentiation of a Laplace transform corresponds to multiplication of the original function by .

This formula shows that differentiation of a Laplace transform of order corresponds to multiplication of the original function by .

Product

This formula represents the Laplace transform of a product and through the contour integral along a vertical line from the corresponding product of Laplace transforms.

Derivative

This time differentiation relation gives the representation for the Laplace transform of the first derivative.

This time differentiation relation gives the representation of the Laplace transform of the derivative.

Integral

This formula shows that the Laplace transform of an indefinite integral gives the product of the reciprocal function of by the Laplace transform of the function.

This formula shows that the Laplace transform of the repeated indefinite integral gives the product of the power function on Laplace transform.

Convolution

The convolution theorem or convolution (Faltung) theorem for the Laplace transform shows that the Laplace transform of a convolution is equal to the product of Laplace transforms of the convoluted functions.

Limit at infinity

The initial value theorem shows that limit at infinity of the Laplace transform multiplied by is the one‐sided limit of the initial function at zero.

Limit at zero

The final value theorem shows that the limit at zero of the Laplace transform multiplied by is the limit of the initial function at infinity.

Sum

Relations with other integral transforms

With inverse Laplace transform

This formula reflects the relation between the direct and the inverse Laplace transforms.

With exponential Fourier transform

This formula shows how the Laplace transform can be represented through the exponential Fourier transform.

With Mellin transform

This formula shows how the Laplace transform can be represented through the Mellin transform.

With Z-transform

This formula shows how the Laplace transform can be represented through the Z‐transform.

Inverse Laplace transform

Definition

This formula is the definition of the inverse Laplace integral transform of the function with respect to the variable .

This formula is the Post–Widder form of the inverse Laplace integral transform of the function with respect to the variable .

Multiple Laplace transform

Definition

This formula is the definition of the double Laplace transform of the function with respect to the variables , .

This formula is the definition of multiple Laplace transforms of the function with respect to the variables , ,…, over .

Mellin transform

Definition

This formula is the definition of the Mellin transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions. Usually, the integral converges in the strip , where and depend on the function and can assume the values . For example, .

General properties

Linearity

This formula reflects the linearity of the Mellin transform.

Scaling

The operation reflects the scaling of the original variable by a positive number in a Mellin transform.

Power

This formula reflects the Mellin transform of the function .

The operation provides the Mellin transform of the original variable raised to a real power .

Shifting

The shift theorem gives the Mellin transform of a product of the original function by some power of .

The operation gives the Mellin transform of a product of the original function by a power of .

Derivative

This formula shows that the Mellin transform of an derivative gives the product of a polynomial and the Mellin transform of the function.

This formula shows that the Mellin transform of gives the product of a power function and the Mellin transform.

This formula shows that the Mellin transform of gives the product of the power function and the Mellin transform.

This formula shows that the Mellin transform of gives the product of a polynomial and the Mellin transform.

Integral

This formula shows that the Mellin transform of an indefinite integral gives the product of and the Mellin transform in the shifted point.

This formula shows how the Mellin transform of a repeated indefinite integral gives the product of a rational function and the Mellin transform in the shifted point.

This formula shows that the Mellin transform of the indefinite integral with a variable lower limit gives the product of and the Mellin transform in the shift point.

This formula demonstrates how the Mellin transform of a repeated indefinite integral with variable lower limits gives the product of a rational function amd the Mellin transform in the shifted point.

Convolution

The Mellin convolution theorem shows that the Mellin transform of a Mellin convolution equals the product of the Mellin transforms.

The generalized Mellin convolution theorem shows that the Mellin transform of the generalized Mellin convolution is equal to the product of the Mellin transforms.

Parseval

This formula is called Mellin–Parseval’s formula.

This formula is called Parseval’s formula.

This representation can be used for evaluation of the general class of integrals from products of Meijer G functions.

Relations with other integral transforms

With inverse Mellin transform

This formula reflects the relation between the direct and the inverse Mellin transforms. The following theorem holds: if an analytical function satisfies the restriction in the strip with some constant , then the integral is a continuous function of the variable and is its Mellin transform in this strip.

With exponential Fourier transform

This formula shows how the Mellin transform can be represented through the exponential Fourier transform.

With Fourier cosine and sine transforms

This formula shows how the Mellin transform can be represented through the cosine and the sine Fourier transforms from the even and odd parts of the function.

With Mellin transform

This formula shows how the Laplace transform can be represented through the Mellin transform.

Inverse Mellin transform

Definition

This formula is the definition of the inverse Mellin integral transform of the function with respect to the variable . If the integral does not converge, the value of is defined in the sense of generalized functions. The condition on usually has the following form: , which represents a vertical strip of convergence for the integral.

Changing the vertical strip of integration leads to a change in the original function . For example, in the case of gamma function you have and .

Multiple Mellin transform

Definition

This formula is the definition of the double Mellin transform of the function with respect to the variables , .

This formula is the definition of the multiple Mellin transform of the function with respect to the variables , ,…, .

General properties

Convolution

The generalized Mellin convolution theorem shows that the double Mellin transform of a Mellin generalized convolution equals the product of the Mellin transforms in the corresponding points.

The generalized Mellin convolution theorem shows that the multiple Mellin transform of a Mellin generalized convolution equals the product of the Mellin transforms in the corresponding points.

Hankel transform

Definition

This formula is the definition of the Hankel integral transform of the function with respect to the variable . If this integral does not converge, the value of is defined in the sense of generalized functions.

General properties

This formula shows that the inverse Hankel integral transform coincides with the direct Hankel integral transform under the restriction .

Hilbert transform

Definition

This formula is the definition of the Hilbert transform of the function with respect to the variable for real .

Inverse Hilbert transform

Definition

This formula is the definition of the inverse Hilbert transform of the function with respect to the variable for real . It coincides with the direct Hilbert transform multiplied by -1.

Relations with other integral transforms

With Hilbert transform

Z-transform

Definition

This formula is the definition of the Z‐transform of the function with respect to the discrete variable at the complex point .

General properties

Linearity

This formula reflects the linearity of the Z‐transform.

Shifting

This formula reflects the shifting property of the Z‐transform.

This formula reflects the shifting property of the Z‐transform.

Scaling

This formula reflects the scaling property of the Z‐transform.

This formula shows that multiplication of a function by leads to repeated differentiation of the Z‐transform.

This formula shows that multiplication of a function by gives the product of and the derivative of the Z‐transform.

Product

The Z‐transform of a product of and is represented through a contour integral along a simple circle‐type contour encircling the origin counterclockwise. All the singular points of the function are located inside the contour. All the singular points of the function are located outside the contour.

Parseval

The Parseval theorem follows from the previous relation for . The integration is performed along a simple circle‐type contour encircling the origin counterclockwise. All the singular points of the function are located inside the contour. All the singular points of the function are located outside the contour.

Correlation

The property is called the cross correlation property of the Z‐transform.

Convolution

The convolution theorem for the Z‐transform shows that the Z‐transform of a convolution sum is equal to the product of the corresponding Z‐transforms.

Limit at infinity

The analog of the Riemann‐Lebesgue theorem shows that the Z‐transform at infinity tends to the initial value .

This formula shows that the Z‐transform at infinity behaves as .

Limit at one

This formula shows that the expression near point behaves as .

Derivative by parameter

This formula reflects the differentiation by parameter property of the Z‐transform.

Limit by parameter

This formula reflects the evaluation limit by parameter property of the Z‐transform.

Integration by parameter

This formula reflects the integration by parameter property of the Z‐transform.

Relations with other integral transforms

With inverse Z-transforms

This formula reflects the relation between the direct and the inverse Z‐transforms.

Inverse Z-transform

Definition

This formula is the definition of the inverse Z‐transform of the function with respect to the variable at the discrete point . The contour integral is performed along a simple circle‐type contour encircling the origin counterclockwise.