Mathematical functions in the Wolfram Language are given names according to definite rules. As with most Wolfram Language functions, the names are usually complete English words, fully spelled out. For a few very common functions, the Wolfram Language uses the traditional abbreviations. Thus the modulo function, for example, is Mod, not Modulo.

Mathematical functions that are usually referred to by a person's name have names in the Wolfram Language of the form PersonSymbol. Thus, for example, the Legendre polynomials are denoted LegendreP[n,x]. Although this convention does lead to longer function names, it avoids any ambiguity or confusion.

When the standard notation for a mathematical function involves both subscripts and superscripts, the subscripts are given before the superscripts in the Wolfram Language form. Thus, for example, the associated Legendre polynomials are denoted LegendreP[n,m,x].

The overall goal of symbolic computation is typically to get formulas that are valid for many possible values of the variables that appear in them. It is however often not practical to try to get formulas that are valid for absolutely every possible value of each variable.

If the Wolfram Language did not automatically replace by 0, then few symbolic computations would get very far. But you should realize that the practical necessity of making such replacements can cause misleading results to be obtained when exceptional values of parameters are used.

The basic operations of the Wolfram Language are nevertheless carefully set up so that whenever possible the results obtained will be valid for almost all values of each variable.

IntegerPart[x] and FractionalPart[x] can be thought of as extracting digits to the left and right of the decimal point. Round[x] is often used for forcing numbers that are close to integers to be exactly integers. Floor[x] and Ceiling[x] often arise in working out how many elements there will be in sequences of numbers with non‐integer spacings.

Boole[expr] is a basic function that turns True and False into 1 and 0. It is sometimes known as the characteristic function or indicator function.

It is often convenient to have functions with different forms in different regions. You can do this using Piecewise.

Piecewise functions appear in systems where there is discrete switching between different domains. They are also at the core of many computational methods, including splines and finite elements. Special cases include such functions as RealAbs, UnitStep, Clip, RealSign, Floor, and Max. The Wolfram Language handles piecewise functions in both symbolic and numerical situations.

The Wolfram Language has three functions for generating pseudorandom numbers that are distributed uniformly over a range of values.

RandomReal and RandomComplex allow you to obtain pseudorandom numbers with any precision.

If you get arrays of pseudorandom numbers repeatedly, you should get a "typical" sequence of numbers, with no particular pattern. There are many ways to use such numbers.

One common way to use pseudorandom numbers is in making numerical tests of hypotheses. For example, if you believe that two symbolic expressions are mathematically equal, you can test this by plugging in "typical" numerical values for symbolic parameters, and then comparing the numerical results. (If you do this, you should be careful about numerical accuracy problems and about functions of complex variables that may not have unique values.)

Other common uses of pseudorandom numbers include simulating probabilistic processes, and sampling large spaces of possibilities. The pseudorandom numbers that the Wolfram Language generates for a range of numbers are always uniformly distributed over the range you specify.

RandomInteger, RandomReal, and RandomComplex are unlike almost any other Wolfram Language functions in that every time you call them, you potentially get a different result. If you use them in a calculation, therefore, you may get different answers on different occasions.

The sequences that you get from RandomInteger, RandomReal, and RandomComplex are not in most senses "truly random", although they should be "random enough" for practical purposes. The sequences are in fact produced by applying a definite mathematical algorithm, starting from a particular "seed". If you give the same seed, then you get the same sequence.

When the Wolfram Language starts up, it takes the time of day (measured in small fractions of a second) as the seed for the pseudorandom number generator. Two different Wolfram Language sessions will therefore almost always give different sequences of pseudorandom numbers.

If you want to make sure that you always get the same sequence of pseudorandom numbers, you can explicitly give a seed for the pseudorandom generator, using SeedRandom.

Every single time RandomInteger, RandomReal, or RandomComplex is called, the internal state of the pseudorandom generator that it uses is changed. This means that subsequent calls to these functions made in subsidiary calculations will have an effect on the numbers returned in your main calculation. To avoid any problems associated with this, you can localize this effect of their use by doing the calculation inside of BlockRandom.

Many applications require random numbers from non‐uniform distributions. The Wolfram Language has many distributions built into the system. You can give a distribution with appropriate parameters instead of a range to RandomInteger or RandomReal.

An additional use of pseudorandom numbers is for selecting from a list. RandomChoice selects with replacement and RandomSample samples without replacement.

Chances are very high that at least one of the choices was repeated in the output. That is because when an element is chosen, it is immediately replaced. On the other hand, if you want to select from an actual set of elements, there should be no replacement.

For any integers a and b, it is always true that b*Quotient[a,b]+Mod[a,b] is equal to a.

Particularly when you are using Mod to get indices for parts of objects, you will often find it convenient to specify an offset.

The greatest common divisor function GCD[n₁,n₂,…] gives the largest integer that divides all the n_i exactly. When you enter a ratio of two integers, the Wolfram Language effectively uses GCD to cancel out common factors and give a rational number in lowest terms.

The least common multiple function LCM[n₁,n₂,…] gives the smallest integer that contains all the factors of each of the n_i.

The Kronecker delta function KroneckerDelta[n₁,n₂,…] is equal to 1 if all the n_i are equal, and is 0 otherwise. can be thought of as a totally symmetric tensor.

You should realize that according to current mathematical thinking, integer factoring is a fundamentally difficult computational problem. As a result, you can easily type in an integer that the Wolfram Language will not be able to factor in anything short of an astronomical length of time. But as long as the integers you give are less than about 50 digits long, FactorInteger should have no trouble. And in special cases it will be able to deal with much longer integers.

Although the Wolfram Language may not be able to factor a large integer, it can often still test whether or not the integer is a prime. In addition, the Wolfram Language has a fast way of finding the ^th prime number.

Particularly in number theory, it is often more important to know the distribution of primes than their actual values. The function PrimePi[x] gives the number of primes that are less than or equal to .

By default, FactorInteger allows only real integers. But with the option setting GaussianIntegers->True, it also handles Gaussian integers, which are complex numbers with integer real and imaginary parts. Just as it is possible to factor uniquely in terms of real primes, it is also possible to factor uniquely in terms of Gaussian primes. There is nevertheless some potential ambiguity in the choice of Gaussian primes. In the Wolfram Language, they are always chosen to have positive real parts, and non‐negative imaginary parts, except for a possible initial factor of or .

The modular power function PowerMod[a,b,n] gives exactly the same results as Mod[a^b,n] for b>0. PowerMod is much more efficient, however, because it avoids generating the full form of a^b.

You can use PowerMod not only to find positive modular powers, but also to find modular inverses. For negative b, PowerMod[a,b,n] gives, if possible, an integer such that . (Whenever such an integer exists, it is guaranteed to be unique modulo n.) If no such integer exists, the Wolfram Language leaves PowerMod unevaluated.

There are distinct Dirichlet characters for a given modulus k, as labeled by the index j. Different conventions can give different orderings for the possible characters.

The Euler totient function gives the number of integers less than that are relatively prime to . An important relation (Fermat's little theorem) is that for all relatively prime to .

The Möbius function is defined to be if is a product of distinct primes, and if contains a squared factor (other than 1). An important relation is the Möbius inversion formula, which states that if for all , then , where the sums are over all positive integers that divide .

The divisor function is the sum of the ^th powers of the divisors of . The function gives the total number of divisors of , and is variously denoted , and . The function , equal to the sum of the divisors of , is often denoted .

The function DivisorSum[n,form] represents the sum of form[i] for all i that divide n. DivisorSum[n,form,cond] includes only those divisors for which cond[i] gives True.

The Jacobi symbol JacobiSymbol[n,m] reduces to the Legendre symbol when is an odd prime. The Legendre symbol is equal to zero if is divisible by ; otherwise it is equal to if is a quadratic residue modulo the prime , and to if it is not. An integer relatively prime to is said to be a quadratic residue modulo if there exists an integer such that . The full Jacobi symbol is a product of the Legendre symbols for each of the prime factors such that .

The extended GCD ExtendedGCD[n₁,n₂,…] gives a list where is the greatest common divisor of the , and the are integers such that . The extended GCD is important in finding integer solutions to linear Diophantine equations.

The multiplicative order function MultiplicativeOrder[k,n] gives the smallest integer such that . Then is known as the order of modulo . The notation is occasionally used.

The generalized multiplicative order function MultiplicativeOrder[k,n,{r₁,r₂,…}] gives the smallest integer such that for some . MultiplicativeOrder[k,n,{-1,1}] is sometimes known as the suborder function of modulo , denoted . MultiplicativeOrder[k,n,{a}] is sometimes known as the discrete log of with respect to the base modulo .

The Carmichael function or least universal exponent gives the smallest integer such that for all integers relatively prime to .

Continued fractions appear in many number theoretic settings. Rational numbers have terminating continued fraction representations. Quadratic irrational numbers have continued fraction representations that become repetitive.

Continued fraction convergents are often used to approximate irrational numbers by rational ones. Those approximations alternate from above and below, and converge exponentially in the number of terms. Furthermore, a convergent of a simple continued fraction is better than any other rational approximation with denominator less than or equal to .

The lattice reduction function LatticeReduce[{v₁,v₂,…}] is used in several kinds of modern algorithms. The basic idea is to think of the vectors of integers as defining a mathematical lattice. Any vector representing a point in the lattice can be written as a linear combination of the form , where the are integers. For a particular lattice, there are many possible choices of the "basis vectors" . What LatticeReduce does is to find a reduced set of basis vectors for the lattice, with certain special properties.

Notice that in the last example, LatticeReduce replaces vectors that are nearly parallel by vectors that are more perpendicular. In the process, it finds some quite short basis vectors.

For a matrix , HermiteDecomposition gives matrices and such that is unimodular, , and is in reduced row echelon form. In contrast to RowReduce, pivots may be larger than 1 because there are no fractions in the ring of integers. Entries above a pivot are minimized by subtracting appropriate multiples of the pivot row.

Bitwise operations act on integers represented as binary bits. BitAnd[n₁,n₂,…] yields the integer whose binary bit representation has ones at positions where the binary bit representations of all of the n_i have ones. BitOr[n₁,n₂,…] yields the integer with ones at positions where any of the n_i have ones. BitXor[n₁,n₂] yields the integer with ones at positions where n₁ or n₂ but not both have ones. BitXor[n₁,n₂,…] has ones where an odd number of the n_i have ones.

Bitwise operations are used in various combinatorial algorithms. They are also commonly used in manipulating bitfields in low‐level computer languages. In such languages, however, integers normally have a limited number of digits, typically a multiple of 8. Bitwise operations in the Wolfram Language in effect allow integers to have an unlimited number of digits. When an integer is negative, it is taken to be represented in two's complement form, with an infinite sequence of ones on the left. This allows BitNot[n] to be equivalent simply to .

SquareFreeQ[n] checks to see if n has a square prime factor. This is done by computing MoebiusMu[n] and seeing if the result is zero; if it is, then n is not squarefree, otherwise it is. Computing MoebiusMu[n] involves finding the smallest prime factor q of n. If n has a small prime factor (less than or equal to ), this is very fast. Otherwise, FactorInteger is used to find q.

NextPrime[n] finds the smallest prime p such that p>n. The algorithm does a direct search using PrimeQ on the odd numbers greater than n.

For RandomPrime[{min,max}] and RandomPrime[max], a random prime p is obtained by randomly selecting from a prime lookup table if max is small and by a random search of integers in the range if max is large. If no prime exists in the specified range, the input is returned unevaluated with an error message.

The algorithm for PrimePowerQ involves first computing the least prime factor p of n and then attempting division by p until either 1 is obtained, in which case n is a prime power, or until division is no longer possible, in which case n is not a prime power.

The Chinese remainder theorem states that a certain class of simultaneous congruences always has a solution. ChineseRemainder[list₁,list₂] finds the smallest non‐negative integer r such that Mod[r,list₂] is list₁. The solution is unique modulo the least common multiple of the elements of list₂.

PrimitiveRoot[n] returns a generator for the group of numbers relatively prime to n under multiplication . This has a generator if and only if n is 2, 4, a power of an odd prime, or twice a power of an odd prime. If n is a prime or prime power, the least positive primitive root will be returned.

The factorial function gives the number of ways of ordering objects. For non‐integer , the numerical value of is obtained from the gamma function, discussed in "Special Functions".

The binomial coefficient Binomial[n,m] can be written as . It gives the number of ways of choosing objects from a collection of objects, without regard to order. The Catalan numbers, which appear in various tree enumeration problems, are given in terms of binomial coefficients as .

The subfactorial Subfactorial[n] gives the number of permutations of objects that leave no object fixed. Such a permutation is called a derangement. The subfactorial is given by .

The multinomial coefficient Multinomial[n₁,n₂,…], denoted , gives the number of ways of partitioning distinct objects into sets of sizes (with ).

The Fibonacci numbers Fibonacci[n] satisfy the recurrence relation with . They appear in a wide range of discrete mathematical problems. For large , approaches the golden ratio. The Lucas numbers LucasL[n] satisfy the same recurrence relation as the Fibonacci numbers do, but with initial conditions and .

The Fibonacci polynomials Fibonacci[n,x] appear as the coefficients of in the expansion of .

The harmonic numbers HarmonicNumber[n] are given by ; the harmonic numbers of order HarmonicNumber[n,r] are given by . Harmonic numbers appear in many combinatorial estimation problems, often playing the role of discrete analogs of logarithms.

The Bernoulli polynomials BernoulliB[n,x] satisfy the generating function relation . The Bernoulli numbers BernoulliB[n] are given by . The appear as the coefficients of the terms in the Euler–Maclaurin summation formula for approximating integrals. The Bernoulli numbers are related to the Genocchi numbers by .

Numerical values for Bernoulli numbers are needed in many numerical algorithms. You can always get these numerical values by first finding exact rational results using BernoulliB[n], and then applying N.

The Euler polynomials EulerE[n,x] have generating function , and the Euler numbers EulerE[n] are given by .

The Nörlund polynomials NorlundB[n,a] satisfy the generating function relation . The Nörlund polynomials give the Bernoulli numbers when . For other positive integer values of , the Nörlund polynomials give higher-order Bernoulli numbers. The generalized Bernoulli polynomials NorlundB[n,a,x] satisfy the generating function relation .

Stirling numbers show up in many combinatorial enumeration problems. For Stirling numbers of the first kind StirlingS1[n,m], gives the number of permutations of elements which contain exactly cycles. These Stirling numbers satisfy the generating function relation . Note that some definitions of the differ by a factor from what is used in the Wolfram Language.

Stirling numbers of the second kind StirlingS2[n,m], sometimes denoted , give the number of ways of partitioning a set of elements into non‐empty subsets. They satisfy the relation .

The Bell numbers BellB[n] give the total number of ways that a set of elements can be partitioned into non‐empty subsets. The Bell polynomials BellB[n,x] satisfy the generating function relation .

The partition function PartitionsP[n] gives the number of ways of writing the integer as a sum of positive integers, without regard to order. PartitionsQ[n] gives the number of ways of writing as a sum of positive integers, with the constraint that all the integers in each sum are distinct.

IntegerPartitions[n] gives a list of the partitions of , with length PartitionsP[n].

Most of the functions here allow you to count various kinds of combinatorial objects. Functions like IntegerPartitions and Permutations allow you instead to generate lists of various combinations of elements.

The signature function Signature[{i₁,i₂,…}] gives the signature of a permutation. It is equal to for even permutations (composed of an even number of transpositions), and to for odd permutations. The signature function can be thought of as a totally antisymmetric tensor, Levi‐Civita symbol or epsilon symbol.

Clebsch–Gordan coefficients and ‐j symbols arise in the study of angular momenta in quantum mechanics, and in other applications of the rotation group. The Clebsch–Gordan coefficients ClebschGordan[{j₁,m₁},{j₂,m₂},{j,m}] give the coefficients in the expansion of the quantum mechanical angular momentum state in terms of products of states .

The 3‐j symbols or Wigner coefficients ThreeJSymbol[{j₁,m₁},{j₂,m₂},{j₃,m₃}] are a more symmetrical form of Clebsch–Gordan coefficients. In the Wolfram Language, the Clebsch–Gordan coefficients are given in terms of 3‐j symbols by .

The 6‐j symbols SixJSymbol[{j₁,j₂,j₃},{j₄,j₅,j₆}] give the couplings of three quantum mechanical angular momentum states. The Racah coefficients are related by a phase to the 6‐j symbols.

The haversine function Haversine[z] is defined by . The inverse haversine function InverseHaversine[z] is defined by . The Gudermannian function Gudermannian[z] is defined as . The inverse Gudermannian function InverseGudermannian[z] is defined by . The Gudermannian satisfies such relations as . The sinc function Sinc[z] is the Fourier transform of a square signal.

There are a number of additional trigonometric and hyperbolic functions that are sometimes used. The versine function is sometimes encountered in the literature and simply is . The coversine function is defined as . The complex exponential is sometimes written as .

When you ask for the square root of a number , you are effectively asking for the solution to the equation . This equation, however, in general has two different solutions. Both and are, for example, solutions to the equation . When you evaluate the "function" , however, you usually want to get a single number, and so you have to choose one of these two solutions. A standard choice is that should be positive for . This is what the Wolfram Language function Sqrt[x] does.

The need to make one choice from two solutions means that Sqrt[x] cannot be a true inverse function for x^2. Taking a number, squaring it, and then taking the square root can give you a different number than you started with.

When you evaluate , there are again two possible answers: and . In this case, however, it is less clear which one to choose.

There is in fact no way to choose so that it is continuous for all complex values of . There has to be a "branch cut"—a line in the complex plane across which the function is discontinuous. The Wolfram Language adopts the usual convention of taking the branch cut for to be along the negative real axis.

When you find an ^th root using , there are, in principle, possible results. To get a single value, you have to choose a particular principal root. There is absolutely no guarantee that taking the ^th root of an ^th power will leave you with the same number.

There are many mathematical functions which, like roots, essentially give solutions to equations. The logarithm function and the inverse trigonometric functions are examples. In almost all cases, there are many possible solutions to the equations. Unique "principal" values nevertheless have to be chosen for the functions. The choices cannot be made continuous over the whole complex plane. Instead, lines of discontinuity, or branch cuts, must occur. The positions of these branch cuts are often quite arbitrary. The Wolfram Language makes the most standard mathematical choices for them.

Euler's constant EulerGamma is given by the limit . It appears in many integrals, and asymptotic formulas. It is sometimes known as the Euler–Mascheroni constant, and denoted .

Catalan's constant Catalan is given by the sum . It often appears in asymptotic estimates of combinatorial functions. It is variously denoted by , , or .

Khinchin's constant Khinchin (sometimes called Khintchine's constant) is given by . It gives the geometric mean of the terms in the continued fraction representation for a typical real number.

Glaisher's constant Glaisher (sometimes called the Glaisher–Kinkelin constant) satisfies , where is the Riemann zeta function. It appears in various sums and integrals, particularly those involving gamma and zeta functions.

Legendre polynomials LegendreP[n,x] arise in studies of systems with three‐dimensional spherical symmetry. They satisfy the differential equation , and the orthogonality relation for .

The associated Legendre polynomials LegendreP[n,m,x] are obtained from derivatives of the Legendre polynomials according to . Notice that for odd integers , the contain powers of , and are therefore not strictly polynomials. The reduce to when .

The spherical harmonics SphericalHarmonicY[l,m,θ,ϕ] are related to associated Legendre polynomials. They satisfy the orthogonality relation for or , where represents integration over the surface of the unit sphere.

Gegenbauer polynomials GegenbauerC[n,m,x] can be viewed as generalizations of the Legendre polynomials to systems with ‐dimensional spherical symmetry. They are sometimes known as ultraspherical polynomials.

GegenbauerC[n,0,x] is always equal to zero. GegenbauerC[n,x] is however given by the limit . This form is sometimes denoted .

Series of Chebyshev polynomials are often used in making numerical approximations to functions. The Chebyshev polynomials of the first kind ChebyshevT[n,x] are defined by . They are normalized so that . They satisfy the orthogonality relation for . The also satisfy an orthogonality relation under summation at discrete points in corresponding to the roots of .

The Chebyshev polynomials of the second kind ChebyshevU[n,z] are defined by . With this definition, . The satisfy the orthogonality relation for .

The name "Chebyshev" is a transliteration from the Cyrillic alphabet; several other spellings, such as "Tschebyscheff", are sometimes used.

Hermite polynomials HermiteH[n,x] arise as the quantum‐mechanical wave functions for a harmonic oscillator. They satisfy the differential equation , and the orthogonality relation for . An alternative form of Hermite polynomials sometimes used is (a different overall normalization of the is also sometimes used).

The Hermite polynomials are related to the parabolic cylinder functions or Weber functions by .

Generalized Laguerre polynomials LaguerreL[n,a,x] are related to hydrogen atom wave functions in quantum mechanics. They satisfy the differential equation , and the orthogonality relation for . The Laguerre polynomials LaguerreL[n,x] correspond to the special case .

Zernike radial polynomials ZernikeR[n,m,x] are used in studies of aberrations in optics. They satisfy the orthogonality relation for .

Jacobi polynomials JacobiP[n,a,b,x] occur in studies of the rotation group, particularly in quantum mechanics. They satisfy the orthogonality relation for . Legendre, Gegenbauer, Chebyshev and Zernike polynomials can all be viewed as special cases of Jacobi polynomials. The Jacobi polynomials are sometimes given in the alternative form .

The Wolfram System includes all the common special functions of mathematical physics found in standard handbooks. Each of the various classes of functions is discussed in turn.

One point you should realize is that in the technical literature there are often several conflicting definitions of any particular special function. When you use a special function in the Wolfram System, therefore, you should be sure to look at the definition given here to confirm that it is exactly what you want.

Special functions in the Wolfram System can usually be evaluated for arbitrary complex values of their arguments. Often, however, the defining relations given in this tutorial apply only for some special choices of arguments. In these cases, the full function corresponds to a suitable extension or analytic continuation of these defining relations. Thus, for example, integral representations of functions are valid only when the integral exists, but the functions themselves can usually be defined elsewhere by analytic continuation.

As a simple example of how the domain of a function can be extended, consider the function represented by the sum . This sum converges only when . Nevertheless, it is easy to show analytically that for any , the complete function is equal to . Using this form, you can easily find a value of the function for any , at least so long as .

The Euler gamma function Gamma[z] is defined by the integral . For positive integer , . can be viewed as a generalization of the factorial function, valid for complex arguments .

There are some computations, particularly in number theory, where the logarithm of the gamma function often appears. For positive real arguments, you can evaluate this simply as Log[Gamma[z]]. For complex arguments, however, this form yields spurious discontinuities. The Wolfram System therefore includes the separate function LogGamma[z], which yields the logarithm of the gamma function with a single branch cut along the negative real axis.

The Euler beta function Beta[a,b] is .

The Pochhammer symbol or rising factorial Pochhammer[a,n] is . It often appears in series expansions for hypergeometric functions. Note that the Pochhammer symbol has a definite value even when the gamma functions that appear in its definition are infinite.

The incomplete gamma function Gamma[a,z] is defined by the integral . The Wolfram System includes a generalized incomplete gamma function Gamma[a,z₀,z₁] defined as .

The alternative incomplete gamma function can therefore be obtained in the Wolfram System as Gamma[a,0,z].

The incomplete beta function Beta[z,a,b] is given by . Notice that in the incomplete beta function, the parameter is an upper limit of integration, and appears as the first argument of the function. In the incomplete gamma function, on the other hand, is a lower limit of integration, and appears as the second argument of the function.

In certain cases, it is convenient not to compute the incomplete beta and gamma functions on their own, but instead to compute regularized forms in which these functions are divided by complete beta and gamma functions. The Wolfram System includes the regularized incomplete beta function BetaRegularized[z,a,b] defined for most arguments by , but taking into account singular cases. The Wolfram System also includes the regularized incomplete gamma function GammaRegularized[a,z] defined by , with singular cases taken into account.

The incomplete beta and gamma functions, and their inverses, are common in statistics. The inverse beta function InverseBetaRegularized[s,a,b] is the solution for in . The inverse gamma function InverseGammaRegularized[a,s] is similarly the solution for in .

Derivatives of the gamma function often appear in summing rational series. The digamma function PolyGamma[z] is the logarithmic derivative of the gamma function, given by . For integer arguments, the digamma function satisfies the relation , where is Euler's constant (EulerGamma in the Wolfram System) and are the harmonic numbers.

The polygamma functions PolyGamma[n,z] are given by . Notice that the digamma function corresponds to . The general form is the ^th, not the ^th, logarithmic derivative of the gamma function. The polygamma functions satisfy the relation . PolyGamma[ν,z] is defined for arbitrary complex by fractional calculus analytic continuation.

BarnesG[z] is a generalization of the Gamma function and is defined by its functional identity BarnesG[z+1]=Gamma[z] BarnesG[z], where the third derivative of the logarithm of BarnesG is positive for positive z. BarnesG is an entire function in the complex plane.

LogBarnesG[z] is a holomorphic function with a branch cut along the negative real-axis such that Exp[LogBarnesG[z]]=BarnesG[z].

Hyperfactorial[n] is a generalization of to the complex plane.

The Dirichlet-L function DirichletL[k,j,s] is implemented as (for ) where is a Dirichlet character with modulus and index .

The Riemann zeta function Zeta[s] is defined by the relation (for ). Zeta functions with integer arguments arise in evaluating various sums and integrals. The Wolfram System gives exact results when possible for zeta functions with integer arguments.

There is an analytic continuation of for arbitrary complex . The zeta function for complex arguments is central to number theoretic studies of the distribution of primes. Of particular importance are the values on the critical line .

In studying , it is often convenient to define the two Riemann–Siegel functions RiemannSiegelZ[t] and RiemannSiegelTheta[t] according to and (for real). Note that the Riemann–Siegel functions are both real as long as is real.

The Stieltjes constants StieltjesGamma[n] are generalizations of Euler's constant that appear in the series expansion of around its pole at ; the coefficient of is . Euler's constant is .

The generalized Riemann zeta function Zeta[s,a] is implemented as , where any term with is excluded.

The Hurwitz zeta function HurwitzZeta[s,a] is implemented as .

The Ramanujan Dirichlet L-function RamanujanTauL[s] is defined by L(s) (for ), with coefficients RamanujanTau[n]. In analogy with the Riemann zeta function, it is again convenient to define the functions RamanujanTauZ[t] and RamanujanTauTheta[t].

The polylogarithm functions PolyLog[n,z] are given by . The polylogarithm function is sometimes known as Jonquière's function. The dilogarithm PolyLog[2,z] satisfies . Sometimes is known as Spence's integral. The Nielsen generalized polylogarithm functions or hyperlogarithms PolyLog[n,p,z] are given by . Polylogarithm functions appear in Feynman diagram integrals in elementary particle physics, as well as in algebraic K‐theory.

The Lerch transcendent LerchPhi[z,s,a] is a generalization of the zeta and polylogarithm functions, given by , where any term with is excluded. Many sums of reciprocal powers can be expressed in terms of the Lerch transcendent. For example, the Catalan beta function can be obtained as .

The Lerch transcendent is related to integrals of the Fermi–Dirac distribution in statistical mechanics by .

The Lerch transcendent can also be used to evaluate Dirichlet L‐series that appear in number theory. The basic L‐series has the form , where the "character" is an integer function with period . L‐series of this kind can be written as sums of Lerch functions with a power of .

LerchPhi[z,s,a,DoublyInfinite->True] gives the doubly infinite sum .

The Hurwitz–Lerch transcendent HurwitzLerchPhi[z,s,a] generalizes HurwitzZeta[s,a] and is defined by .

The Wolfram System has two forms of exponential integral: ExpIntegralE and ExpIntegralEi.

The exponential integral function ExpIntegralE[n,z] is defined by .

The second exponential integral function ExpIntegralEi[z] is defined by (for ), where the principal value of the integral is taken.

The logarithmic integral function LogIntegral[z] is given by (for ), where the principal value of the integral is taken. is central to the study of the distribution of primes in number theory. The logarithmic integral function is sometimes also denoted by . In some number theoretic applications, is defined as , with no principal value taken. This differs from the definition used in the Wolfram System by the constant .

The sine and cosine integral functions SinIntegral[z] and CosIntegral[z] are defined by and . The hyperbolic sine and cosine integral functions SinhIntegral[z] and CoshIntegral[z] are defined by and .

The error function Erf[z] is the integral of the Gaussian distribution, given by . The complementary error function Erfc[z] is given simply by . The imaginary error function Erfi[z] is given by . The generalized error function Erf[z₀,z₁] is defined by the integral . The error function is central to many calculations in statistics.

The inverse error function InverseErf[s] is defined as the solution for in the equation . The inverse error function appears in computing confidence intervals in statistics as well as in some algorithms for generating Gaussian random numbers.

Closely related to the error function are the Fresnel integrals FresnelC[z] defined by and FresnelS[z] defined by . Fresnel integrals occur in diffraction theory.

The Bessel functions BesselJ[n,z] and BesselY[n,z] are linearly independent solutions to the differential equation . For integer , the are regular at , while the have a logarithmic divergence at .

Bessel functions arise in solving differential equations for systems with cylindrical symmetry.

is often called the Bessel function of the first kind, or simply the Bessel function. is referred to as the Bessel function of the second kind, the Weber function, or the Neumann function (denoted ).

The Hankel functions (or Bessel functions of the third kind) HankelH1[n,z] and HankelH2[n,z] give an alternative pair of solutions to the Bessel differential equation, related according to .

The spherical Bessel functions SphericalBesselJ[n,z] and SphericalBesselY[n,z], as well as the spherical Hankel functions SphericalHankelH1[n,z] and SphericalHankelH2[n,z], arise in studying wave phenomena with spherical symmetry. These are related to the ordinary functions by , where and can be and , and , or and . For integer , spherical Bessel functions can be expanded in terms of elementary functions by using FunctionExpand.

The modified Bessel functions BesselI[n,z] and BesselK[n,z] are solutions to the differential equation . For integer , is regular at ; always has a logarithmic divergence at . The are sometimes known as hyperbolic Bessel functions.