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The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.The (unilateral) Laplace transform (not to be confused with the Lie derivative, also commonly denoted ) is defined by(1)where is defined for (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a bilateral Laplace transform is sometimes also defined as(2)(Oppenheim et al. 1997). The unilateral Laplace transform is implemented in the Wolfram Language as LaplaceTransform[f[t], t, s] and the inverse Laplace transform as InverseRadonTransform.The inverse Laplace transform is known as the Bromwich integral, sometimes known as the Fourier-Mellin integral..

The integral transformwhere is a modified Bessel function of the second kind. Note the lower limit of 0, not as implied in Samko et al. (1993, p. 23, eqn. 1.101).

The integral transformwhere is the gamma function, is a hypergeometric function, where denotes the truncated power function. Note the lower limit of 0, not as implied in Samko et al. (1993, p. 23, eqn. 1.101).

If there are two functions and with the same integral transform(1)then a null function can be defined by(2)so that the integral(3)vanishes for all .

whereThis result was originally derived using harmonicanalysis, but also follows from a wavelets viewpoint.

The integral transform defined bywhere is the truncated power function and is an associated Legendre polynomial. Note the lower limit of 0, not as implied in Samko et al. (1993, p. 23, eqn. 1.101).

A one-sided (singly infinite) Laplace transform,This is the most common variety of Laplace transform and it what is usually meant by "the" Laplace transform. The unilateral Laplace transform is implemented in the Wolfram Language as LaplaceTransform[expr, t, s].

The inverse of the Laplace transform, givenbywhere is a vertical contour in the complex plane chosen so that all singularities of are to the left of it.

The Hilbert transform (and its inverse) are the integraltransform(1)(2)where the Cauchy principal value is taken in each of the integrals. The Hilbert transform is an improper integral.In the following table, is the rectangle function, is the sinc function, is the delta function, and are impulse symbols, and is a confluent hypergeometric function of the first kind.

The Hartley Transform is an integral transform which shares some features with the Fourier transform, but which, in the most common convention, multiplies the integral kernel by(1)instead of by , giving the transform pair(2)(3)(Bracewell 1986, p. 10, Bracewell 1999, p. 179).The Hartley transform produces real output for a real input, and is its own inverse. It therefore can have computational advantages over the discrete Fourier transform, although analytic expressions are usually more complicated for the Hartley transform.In the discrete case, the kernel is multiplied by(4)instead of(5)The discrete version of the Hartley transform--using an alternate convention withthe plus sign replaced by a minus sine can be written explicitly as(6)(7)where denotes the Fourier transform. The Hartley transform obeys the convolution property(8)where(9)(10)(11)Like the fast Fourier transform, there is a "fast"..

A two-sided (doubly infinite) Laplace transform,While some authors use this as the primary definition of "the" Laplace transform (Oppenheim et al. 1997), it is much more common for the unilateral Laplace transform to be considered the primary definition.

Suppose that in some neighborhood of ,(1)for some function (say analytic or integrable) . Then(2)These functions form a forward/inverse transform pair. For example, taking for all gives(3)and(4)which is simply the usual integral formula for the gammafunction.Ramanujan used this theorem to generate amazing identities by substituting particular values for .

The operator defined byfor , where is the unit open disk and is the complex conjugate (Hedenmalm et al. 2000, p. 29).

Let be any function, say analytic or integrable. Then(1)and(2)where is the Dirichlet lambda function and is the gamma function. Equation (◇) is obtained from (◇) by defining(3)These formulas give valid results only for certain classes of functions, and are connected with Mellin transforms (Hardy 1999, p. 15).

Let be a periodic sequence, then the autocorrelation of the sequence, sometimes called the periodic autocorrelation (Zwillinger 1995, p. 223), is the sequence(1)where denotes the complex conjugate and the final subscript is understood to be taken modulo .Similarly, for a periodic array with and , the autocorrelation is the -dimensional matrix given by(2)where the final subscripts are understood to be taken modulo and , respectively.For a complex function , the autocorrelation is defined by(3)(4)where denotes cross-correlation and is the complex conjugate (Bracewell 1965, pp. 40-41).Note that the notation is sometimes used for and that the quantity(5)is sometimes also known as the autocorrelation of a continuous real function (Papoulis 1962, p. 241).The autocorrelation discards phase information, returning only the power, and is therefore an irreversible operation.There is also a somewhat surprising and..

Define(1)and(2)for a nonnegative integer and .So, for example, the first few values of are(3)(4)(5)(6)(7)(8)(9)Then a function can be written as a series expansion by(10)The functions and are all orthogonal in , with(11)(12)for in the first case and in the second.These functions can be used to define wavelets. Let a function be defined on intervals, with a power of 2. Then an arbitrary function can be considered as an -vector , and the coefficients in the expansion can be determined by solving the matrix equation(13)for , where is the matrix of basis functions. For example, the fourth-order Haar function wavelet matrix is given by(14)(15)

The following integral transform relationship, known as the Abel transform, exists between two functions and for ,(1)(2)(3)The Abel transform is used in calculating the radial mass distribution of galaxies (Binney and Tremaine 1987, p. 651; Arfken and Weber 2005, p. 1014) and inverting planetary radio occultation data to obtain atmospheric information as a function of height.Bracewell (1999, p. 262) defines a slightly different form of the Abel transform given by(4)The following table gives a number of common Abel transform pairs (Bracewell 1999, p. 264). Here,(5)where is the rectangle function, and(6)(7)where is a Bessel function of the first kind and is a Struve function.conditions

The Mellin transform is the integral transformdefined by(1)(2)It is implemented in the Wolfram Language as MellinTransform[expr, x, s]. The transform exists if the integral(3)is bounded for some , in which case the inverse exists with . The functions and are called a Mellin transform pair, and either can be computed if the other is known.The following table gives Mellin transforms of common functions (Bracewell 1999, p. 255). Here, is the delta function, is the Heaviside step function, is the gamma function, is the incomplete beta function, is the complementary error function erfc, and is the sine integral.convergenceAnother example of a Mellin transform is the relationship between the Riemann function and the Riemann zeta function ,(4)(5)A related pair is used in one proof of the primenumber theorem (Titchmarsh 1987, pp. 51-54 and equation 3.7.2)...

A unit circle is a circle of unit radius, i.e., of radius 1.The unit circle plays a significant role in a number of different areas of mathematics. For example, the functions of trigonometry are most simply defined using the unit circle. As shown in the figure above, a point on the terminal side of an angle in angle standard position measured along an arc of the unit circle has as its coordinates so that is the horizontal coordinate of and is its vertical component.As a result of this definition, the trigonometric functions are periodic with period .Another immediate result of this definition is the ability to explicitly write the coordinates of a number of points lying on the unit circle with very little computation. In the figure above, for example, points , , , and correspond to angles of , , , and radians, respectively, whereby it follows that , , , and . Similarly, this method can be used to find trigonometric values associated to integer multiples of..

The matrix product of a square set of data and a matrix of basis vectors consisting of Walsh functions. By taking advantage of the nested structure of the natural ordering of the Walsh functions, it is possible to speed the transform up from to steps, resulting in the so-called fast Walsh transform (Wolfram 2002, p. 1073). Walsh transforms are widely used for signal and image processing, and can also be used for image compression (Wolfram 2002, p. 1073).

Expresses a function in terms of its Radon transform,(1)(2)

The transformation of a sequence into a sequence by the formula(1)where is a Stirling number of the second kind. The inverse transform is given by(2)where is a Stirling number of the first kind (Sloane and Plouffe 1995, p. 23).The following table summarized Stirling transforms for some common sequences, where denotes the Iverson bracket and denotes the primes.OEIS1A0001101, 1, 2, 5, 15, 52, 203, ...A0054930, 1, 3, 10, 37, 151, 674, ...A0001101, 2, 5, 15, 52, 203, 877, ...A0855070, 0, 1, 4, 13, 41, 136, 505, ...A0244301, 0, 1, 3, 8, 25, 97, 434, 2095, ...A0244290, 1, 1, 2, 7, 27, 106, 443, ...A0339991, , 1, , 1, , ...Here, gives the Bell numbers. has the exponential generating function(3)

The Hankel transform (of order zero) is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel and also called the Fourier-Bessel transform. It is defined as(1)(2)Let(3)(4)so that(5)(6)(7)(8)(9)(10)Then(11)(12)(13)(14)(15)(16)where is a zeroth order Bessel function of the first kind.Therefore, the Hankel transform pairs are(17)(18)A slightly differently normalized Hankel transform and its inverse are implemented in the Wolfram Language as HankelTransform[expr, r, s] and InverseHankelTransform[expr, s, r], respectively.The following table gives Hankel transforms for a number of common functions (Bracewell 1999, p. 249). Here, is a Bessel function of the first kind and is a rectangle function equal to 1 for and 0 otherwise, and(19)(20)where is a Bessel function of the first kind, is a Struve function and is a modified Struve function.1The Hankel transform..

There are two sorts of transforms known as the fractional Fourier transform.The linear fractional Fourier transform is a discrete Fourier transform in which the exponent is modified by the addition of a factor ,However, such transforms may not be consistent with their inverses unless is an integer relatively prime to so that . Fractional fourier transforms are implemented in the Wolfram Language as Fourier[list, FourierParameters -> a, b], where is an additional scaling parameter. For example, the plots above show 2-dimensional fractional Fourier transforms of the function for parameter ranging from 1 to 6.The quadratic fractional Fourier transform is defined in signal processing and optics. Here, the fractional powers of the ordinary Fourier transform operation correspond to rotation by angles in the time-frequency or space-frequency plane (phase space). So-called fractional Fourier domains correspond to oblique axes in..

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