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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..

(1)where(2)and(3)is the delta function.(4)(5)(6)(7)(8)(9)(10)(11)From Gradshteyn and Ryzhik (2000, equation 3.741.3),(12)so(13)This can also be written explicitly in the form(14)

For a delta function at ,(1)(2)(3)(4)(5)

Let the two-dimensional cylinder function be defined by(1)Then the Radon transform is given by(2)where(3)is the delta function. Rewriting in polarcoordinates then gives(4)Now use the harmonic addition theoremto write(5)with a phase shift. Then(6)(7)(8)Then use(9)which, with , becomes(10)Define(11)(12)(13)so the inner integral is(14)(15)and the Radon transform becomes(16)(17)(18)Converting to using ,(19)(20)(21)which could have been derived more simply by(22)

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 Fourier transform of the Heaviside step function is given by(1)(2)where is the delta function.

Plancherel's theorem states that the integral of the squared modulus of a function is equal to the integral of the squared modulus of its spectrum. It corresponds to Parseval's theorem for Fourier series. It is sometimes also known as Rayleigh's theory, since it was first used by Rayleigh (1889) in the investigation of blackbody radiation. In 1910, Plancherel first established conditions under which the theorem holds (Titchmarsh 1924; Bracewell 1965, p. 113).In other words, let be a function that is sufficiently smooth and that decays sufficiently quickly near infinity so that its integrals exist. Further, let and be Fourier transform pairs so that(1)(2)where denotes the complex conjugate.Then(3)(4)(5)(6)(7)(8)(9)where is the delta function.

The Fourier transform of a Gaussian function is given by(1)(2)(3)The second integrand is odd, so integration over a symmetrical range gives 0. The value of the first integral is given by Abramowitz and Stegun (1972, p. 302, equation 7.4.6), so(4)so a Gaussian transforms to another Gaussian.

Let and be the Fourier transforms of and , respectively. Thenwhere denotes the complex conjugate.

The Fourier transform of is given by(1)(2)Now let so , then(3)which, from the damped exponentialcosine integral, gives(4)which is a Lorentzian function.

Simplemindedly, a number theoretic transform is a generalization of a fast Fourier transform obtained by replacing with an th primitive root of unity. This effectively means doing a transform over the quotient ring instead of the complex numbers . The theory is rather elegant and uses the language of finite fields and number theory.

The Fourier transform of the deltafunction is given by(1)(2)

The important property of Fourier transforms that can be expressed in terms of as follows,

(1)(2)(3)where is the delta function.

The Fourier transform of the constant function is given by(1)(2)according to the definition of the delta function.

The Fourier transform is a generalization of the complex Fourier series in the limit as . Replace the discrete with the continuous while letting . Then change the sum to an integral, and the equations become(1)(2)Here,(3)(4)is called the forward () Fourier transform, and(5)(6)is called the inverse () Fourier transform. The notation is introduced in Trott (2004, p. xxxiv), and and are sometimes also used to denote the Fourier transform and inverse Fourier transform, respectively (Krantz 1999, p. 202).Note that some authors (especially physicists) prefer to write the transform in terms of angular frequency instead of the oscillation frequency . However, this destroys the symmetry, resulting in the transform pair(7)(8)(9)(10)To restore the symmetry of the transforms, the convention(11)(12)(13)(14)is sometimes used (Mathews and Walker 1970, p. 102).In general, the Fourier transform pair may be defined using..

The Fourier sine transform is the imaginary partof the full complex Fourier transform,(1)(2)The Fourier sine transform of a function is implemented as FourierSinTransform[f, x, k], and different choices of and can be used by passing the optional FourierParameters -> a, b option. In this work, and .The discrete Fourier sine transform of a list of real numbers can be computed in the Wolfram Language using FourierDST[l].

The square matrix with entries given by(1)for , 1, 2, ..., , where i is the imaginary number , and normalized by to make it a unitary. The Fourier matrix is given by(2)and the matrix by(3)(4)In general,(5)with(6)where is the identity matrix and is the diagonal matrix with entries 1, , ..., . Note that the factorization (which is the basis of the fast Fourier transform) has two copies of in the center factor matrix.

The Fourier cosine transform of a real function is the realpart of the full complex Fourier transform,(1)(2)The Fourier cosine transform of a function is implemented as FourierCosTransform[f, x, k], and different choices of and can be used by passing the optional FourierParameters -> a, b option. In this work, and .The discrete Fourier cosine transform of a list of real numbers can be computed in the Wolfram Language using FourierDCT[l].

(1)(2)(3)where is the delta function.

The fast Fourier transform (FFT) is a discrete Fourier transform algorithm which reduces the number of computations needed for points from to , where lg is the base-2 logarithm. FFTs were first discussed by Cooley and Tukey (1965), although Gauss had actually described the critical factorization step as early as 1805 (Bergland 1969, Strang 1993). A discrete Fourier transform can be computed using an FFT by means of the Danielson-Lanczos lemma if the number of points is a power of two. If the number of points is not a power of two, a transform can be performed on sets of points corresponding to the prime factors of which is slightly degraded in speed. An efficient real Fourier transform algorithm or a fast Hartley transform (Bracewell 1999) gives a further increase in speed by approximately a factor of two. Base-4 and base-8 fast Fourier transforms use optimized code, and can be 20-30% faster than base-2 fast Fourier transforms. prime factorization..

Let be the rectangle function, then the Fourier transform iswhere is the sinc function.

The continuous Fourier transform is definedas(1)(2)Now consider generalization to the case of a discrete function, by letting , where , with , ..., . Writing this out gives the discrete Fourier transform as(3)The inverse transform is then(4)Discrete Fourier transforms (DFTs) are extremely useful because they reveal periodicities in input data as well as the relative strengths of any periodic components. There are however a few subtleties in the interpretation of discrete Fourier transforms. In general, the discrete Fourier transform of a real sequence of numbers will be a sequence of complex numbers of the same length. In particular, if are real, then and are related by(5)for , 1, ..., , where denotes the complex conjugate. This means that the component is always real for real data.As a result of the above relation, a periodic function will contain transformed peaks in not one, but two places. This happens because the periods of the input data..

The structure factor of a discrete set is the Fourier transform of -scatterers of equal strengths on all points of ,

Let be the ramp function, then the Fourier transform of is given by(1)(2)where is the derivative of the delta function.

If is piecewise continuous and has a generalized Fourier series(1)with weighting function , it must be true that(2)(3)But the coefficient of the generalizedFourier series is given by(4)so(5)(6)Equation (6) is an inequality if the functions do not form a complete orthogonal system. If they are a complete orthogonal system, then the inequality (2) becomes an equality, so (6) becomes an equality and is known as Parseval's theorem.If has a simple Fourier series expansion with coefficients , , , and , ..., , then(7)The inequality can also be derived from Schwarz'sinequality(8)by expanding in a superposition of eigenfunctions of , . Then(9)and(10)(11)where is the complex conjugate. If is normalized, then and(12)

The Fourier transform of the generalized function is given by(1)(2)(3)(4)where denotes the Cauchy principal value. Equation (4) can also be written as the single equation(5)where is the Heaviside step function. The integrals follow from the identity(6)(7)(8)

A deconvolution algorithm (sometimes abbreviated MEM) which functions by minimizing a smoothness function ("entropy") in an image. Maximum entropy is also called the all-poles model or autoregressive model. For images with more than a million pixels, maximum entropy is faster than the CLEAN algorithm.MEM is commonly employed in astronomical synthesis imaging. In this application, the resolution depends on the signal-to-noise ratio, which must be specified. Therefore, resolution is image dependent and varies across the map. MEM is also biased, since the ensemble average of the estimated noise is nonzero. However, this bias is much smaller than the noise for pixels with a . It can yield super-resolution, which can usually be trusted to an order of magnitude in solid angle.Two definitions of "entropy" normalized tothe flux in the image are(1)(2)where is a "default image" and is the smoothed image. Several..

The inversion of a convolution equation, i.e., the solution for of an equation of the formgiven and , where is the noise and denotes the convolution. Deconvolution is ill-posed and will usually not have a unique solution even in the absence of noise.Linear deconvolution algorithms include inverse filtering and Wiener filtering. Nonlinear algorithms include the CLEAN algorithm, maximum entropy method, and LUCY.

Let and be arbitrary functions of time with Fourier transforms. Take(1)(2)where denotes the inverse Fourier transform (where the transform pair is defined to have constants and ). Then the convolution is(3)(4)Interchange the order of integration,(5)(6)(7)So, applying a Fourier transform to each side,we have(8)The convolution theorem also takes the alternate forms(9)(10)(11)

A convolution is an integral that expresses the amount of overlap of one function as it is shifted over another function . It therefore "blends" one function with another. For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam (the Fourier transform of the sampling distribution). The convolution is sometimes also known by its German name, faltung ("folding").Convolution is implemented in the Wolfram Language as Convolve[f, g, x, y] and DiscreteConvolve[f, g, n, m].Abstractly, a convolution is defined as a product of functions and that are objects in the algebra of Schwartz functions in . Convolution of two functions and over a finite range is given by(1)where the symbol denotes convolution of and .Convolution is more often taken over an infinite range,(2)(3)(Bracewell 1965, p. 25) with the variable (in this case ) implied, and also occasionally..

The apodization function(1)Its full width at half maximum is .Its instrument function is(2)(3)where is a Bessel function of the first kind. This function has a maximum of . To investigate the instrument function, define the dimensionless parameter and rewrite the instrument function as(4)Finding the full width at half maximumthen amounts to solving(5)which gives , so for , the full width at half maximum is(6)The maximum negative sidelobe of times the peak, and maximum positive sidelobe of 0.356044 times the peak.

An apodization function(1)having instrument function(2)The peak of is . The full width at half maximum of can found by setting to obtain(3)and solving for , yielding(4)Therefore, with ,(5)The extrema are given by taking the derivative of , substituting , and setting equal to 0(6)Solving this numerically gives sidelobes at 0.715148 (), 1.22951 (0.256749), 1.73544 (), ....

An asymmetrical apodization function definedby(1)where the two-sided portion is long (total) and the one-sided portion is long (Schnopper and Thompson 1974, p. 508). The instrument function is(2)

The finite Fourier cosine transform of an apodization function, also known as an apparatus function. The instrument function corresponding to a given apodization function is(1)which, upon expanding the complex exponential,(2)(3)For an even function, the left integrand is even (and hence is equal to twice its value over half its interval) and the right integrand is odd (and hence equal to 0), so(4)

An apodization function, also called the Hann function, frequently used to reduce leakage in discrete Fourier transforms. The illustrations above show the Hanning function, its instrument function, and a blowup of the instrument function sidelobes. It is named after the Austrian meteorologist Julius von Hann (Blackman and Tukey 1959, pp. 98-99). The Hanning function is given by(1)(2)Its full width at half maximum is .It has instrument function(3)(4)To investigate the instrument function, define the dimensionless parameter and rewrite the instrument function as(5)The half-maximum can then be seen to occur at(6)so for , the full width at half maximum is(7)To find the extrema, take the derivative(8)and equate to zero. The first two roots are and 10.7061..., corresponding to the first sidelobe minimum () and maximum (), respectively...

An apodization function chosen to minimize the height of the highest sidelobe (Hamming and Tukey 1949, Blackman and Tukey 1959). The Hamming function is given by(1)and its full width at half maximum is .The corresponding instrument function is(2)This apodization function is close to the one produced by the requirement that the instrument function goes to 0 at . The FWHM is , the peak is 1.08, and the peak negative and positive sidelobes (in units of the peak) are and 0.00734934, respectively.From the apodization function, a general symmetric apodization function can be written as a Fourier series(3)where the coefficients satisfy(4)The corresponding instrument function is(5)To obtain an apodization function with zero at , use(6)so(7)(8)(9)(10)(11)

The apodization functionIts full width at half maximum is .Its instrument function iswhich has a maximum of and full width at half maximum .

The apodization functionIts full width at half maximum is .Its instrument function iswhere is a Bessel function of the first kind. This has a maximum of , and full width at half maximum of .

An apodization function given by(1)which has full width at half maximum of . This function is defined so that the coefficients are approximations in the general expansion(2)to(3)(4)(5)which produce zeros of at and .The corresponding instrument function is(6)where is the sinc function. It is full width at half maximum is .

The apodization function(1)which is a generalization of the one-argument triangle function. Its full width at half maximum is .It has instrument function(2)where is the sinc function. The peak of is , and the full width at half maximum is given by setting and numerically solving(3)for , yielding(4)Therefore, with ,(5)The function is always positive, so there are no negative sidelobes. The extrema are given by differentiating with respect to , defining , and setting equal to 0,(6)Solving this numerically gives minima of 0 at , 2, 3, ..., and sidelobes of 0.047190, 0.01648, 0.00834029, ... at , 2.45892, 3.47089, ....

An apodization function (also called a tapering function or window function) is a function used to smoothly bring a sampled signal down to zero at the edges of the sampled region. This suppresses leakage sidelobes which would otherwise be produced upon performing a discrete Fourier transform, but the suppression is at the expense of widening the lines, resulting in a decrease in the resolution.A number of apodization functions for symmetrical (two-sided) interferograms are summarized below, together with the instrument functions (or apparatus functions) they produce and a blowup of the instrument function sidelobes. The instrument function corresponding to a given apodization function can be computed by taking the finite Fourier cosine transform,(1)typeapodization functioninstrument functionBartlettBlackmanConnescosineGaussianHammingHanninguniform1Welchwhere(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)The following..

The so-called generalized Fourier integral is a pair of integrals--a "lower Fourier integral" and an "upper Fourier integral"--which allow certain complex-valued functions to be decomposed as the sum of integral-defined functions, each of which resembles the usual Fourier integral associated to and maintains several key properties thereof.Let be a real variable, let be a complex variable, and let be a function for which as , for which as , and for which has an analytic Fourier integral where here, are finite real constants. Next, define the upper and lower generalized Fourier integrals and associated to , respectively, by(1)and(2)on the complex regions and , respectively. Then, for and ,(3)where the first integral summand equals for and is zero for while the second integral summand is zero for and equals for . The decomposition () is called the generalized Fourier integral corresponding to .Note that some literature..

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)...

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 triangle function is the function(1)(2)(3)where is the rectangle function, is the Heaviside step function, and denotes convolution. An obvious generalization used as an apodization function goes by the name of the Bartlett function.The piecewise version of the triangle function is implemented in the Wolfram Language as UnitTriangle[x], while the generalized function version is implemented as HeavisideLambda[x].There is also a three-argument function known as the triangle function,(4)It follows that(5)

The Radon transform is an integral transform whose inverse is used to reconstruct images from medical CT scans. A technique for using Radon transforms to reconstruct a map of a planet's polar regions using a spacecraft in a polar orbit has also been devised (Roulston and Muhleman 1997).The Radon and inverse Radon transforms are implemented in the Wolfram Language as RadonTransform and InverseRadonTransform, respectively.The Radon transform can be defined by(1)(2)(3)where is the slope of a line, is its intercept, and is the delta function. The inverse Radon transform is(4)where is a Hilbert transform. The transform can also be defined by(5)where is the perpendicular distance from a line to the origin and is the angle formed by the distance vector.Using the identity(6)where is the Fourier transform, gives the inversion formula(7)The Fourier transform can be eliminated by writing(8)where is a weighting function such as(9)(10)Nievergelt..

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..

The engineering terminology for one use of Fouriertransforms. By breaking up a wave pulse into its frequency spectrum(1)the entire signal can be written as a sum of contributions from each frequency,(2)If the signal is modified in some way, it will become(3)(4)(5)(6)where is known as the "transfer function." Fourier transforming and ,(7)(8)From the convolution theorem,(9)

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