Naming Conventions

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, isMod, notModulo.

Mathematical functions that are usually referred to by a person's name have names in the Wolfram Language of the formPersonSymbol. Thus, for example, the Legendre polynomials are denotedLegendreP[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 givenbefore the superscripts in the Wolfram Language form. Thus, for example, the associated Legendre polynomials are denotedLegendreP[n,m,x].

Generic and Non‐Generic Cases

This gives a result for the integral of that is valid for almost all values of:

For the special case of, however, the correct result is different:

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.

The Wolfram Language always replaces by:

If is equal to 0, however, then the true result is not 0:

This construct treats both cases, but would be quite unwieldy to use:

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.

is not automatically replaced by:

If it were, then the result here would be, which is incorrect:

This makes the assumption that is a positive real variable, and does the replacement:

Numerical Functions

Functions relating real numbers and integers.

Extracting integer and fractional parts.

IntegerPart[x] andFractionalPart[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] andCeiling[x] often arise in working out how many elements there will be in sequences of numbers with non‐integer spacings.

Numerical functions of real variables.

Numerical functions of complex variables.

Piecewise Functions

Turning conditions into numbers.

Boole[expr] is a basic function that turnsTrue andFalse into1 and0. It is sometimes known as thecharacteristic function orindicator function.

This gives the area of a unit disk:

Piecewise functions.

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

This plots a piecewise function:

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 asRealAbs,UnitStep,Clip,RealSign,Floor, andMax. The Wolfram Language handles piecewise functions in both symbolic and numerical situations.

This generates a square wave:

Here is the integral of the square wave:

Pseudorandom Numbers

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

Pseudorandom number generation.

Generating tables of pseudorandom numbers.

This will give 0 or 1 with equal probability:

This gives a pseudorandom complex number:

This gives a list of 10 pseudorandom integers between 0 and 9 (inclusive):

This gives a matrix of pseudorandom reals between 0 and 1:

RandomReal andRandomComplex allow you to obtain pseudorandom numbers with any precision.

Changing the precision for pseudorandom numbers.

Here is a 30‐digit pseudorandom real number in the range 0 to 1:

Here is a list of four 20-digit pseudorandom complex numbers:

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.)

Here is a symbolic equation:

Substituting in a random numerical value shows that the equation is not alwaysTrue:

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, andRandomComplex 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 fromRandomInteger,RandomReal, andRandomComplex 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, usingSeedRandom.

Pseudorandom number generator seed.

This reseeds the pseudorandom generator:

Here are three pseudorandom numbers:

If you reseed the pseudorandom generator with the same seed, you get the same sequence of pseudorandom numbers:

Every single timeRandomInteger,RandomReal, orRandomComplex 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 ofBlockRandom.

Localizing the effects of usingRandomInteger,RandomReal, orRandomComplex.

By localizing the calculation insideBlockRandom, the internal state of the pseudorandom generator is restored after generating the first list:

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 toRandomInteger orRandomReal.

Generating pseudorandom numbers with non-uniform distributions.

This generates 12 integers distributed by the Poisson distribution with mean 3:

This generates a 4×4 matrix of real numbers using the standard normal distribution:

This generates five high-precision real numbers distributed normally with mean 2 and standard deviation 4:

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

Selecting at random.

Choose 10 items at random from the digits 0 through 9:

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.

Sample 10 items at random from the digits 0 through 9 without replacement. The result is a random permutation of the digits:

Sample 10 items from a set having different frequencies for each digit:

Integer and Number Theoretic Functions

Some integer functions.

The remainder on dividing 17 by 3:

The integer part of 17/3:

Mod also works with real numbers:

The result fromMod always has the same sign as the second argument:

For any integersa andb, it is always true thatb*Quotient[a,b]+Mod[a,b] is equal toa.

Integer remainders with offsets.

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

This effectively extracts the 18^th part of the list, with the list treated cyclically:

Thegreatest common divisor functionGCD[n₁,n₂,…] gives the largest integer that divides all then_i exactly. When you enter a ratio of two integers, the Wolfram Language effectively usesGCD to cancel out common factors and give a rational number in lowest terms.

Theleast common multiple functionLCM[n₁,n₂,…] gives the smallest integer that contains all the factors of each of then_i.

The largest integer that divides both 24 and 15 is 3:

TheKronecker delta functionKroneckerDelta[n₁,n₂,…] is equal to 1 if all then_i are equal, and is 0 otherwise. can be thought of as a totally symmetric tensor.

This gives a totally symmetric tensor of rank 3:

Integer factoring and related functions.

This gives the factors of 24 as,. The first element in each list is the factor; the second is its exponent:

Here are the factors of a larger integer:

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.

Here is a rather special long integer:

The Wolfram Language can easily factor this special integer:

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.

It is often much faster to test whether a number is prime than to factor it:

Here is a plot of the first 100 primes:

This is the millionth prime:

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

This gives the number of primes less than a billion:

PrimeNu gives the number of distinct primes in the factorization ofn:

PrimeOmega gives the number of prime factors counting multiplicities inn:

Liouville's function gives where is the number of prime factors counting multiplicity:

The Mangoldt function returns the log of prime power base or zero when composite:

By default,FactorInteger allows only real integers. But with the option settingGaussianIntegers->True, it also handlesGaussian 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.

Over the Gaussian integers, 2 can be factored as:

Here are the factors of a Gaussian integer:

Some functions from number theory.

Themodular power functionPowerMod[a,b,n] gives exactly the same results asMod[a^b,n] forb>0.PowerMod is much more efficient, however, because it avoids generating the full form ofa^b.

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

PowerMod is equivalent to usingPower, thenMod, but is much more efficient:

This gives the modular inverse of 3 modulo 7:

Multiplying the inverse by 3 modulo 7 gives 1, as expected:

This finds the smallest non‐negative integer so that is equal to 3 mod 11:

This verifies the result:

This returns all integers less than 11 which satisfy the relation:

Ifd does not have a square root modulon,PowerMod[d,n] will remain unevaluated andPowerModList will return an empty list:

This checks that 3 is not a square modulo 5:

Even for a large modulus, the square root can be computed fairly quickly:

PowerMod[d,n] also works for composite:

There are distinct Dirichlet characters for a given modulusk, as labeled by the indexj. Different conventions can give different orderings for the possible characters.

DirichletCharacter works for very large numbers:

TheEuler 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.

TheMöbius 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.

Thedivisor 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.

For prime,:

The result is 1, as guaranteed by Fermat's little theorem:

This gives a list of all the divisors of 24:

gives the total number of distinct divisors of 24:

The functionDivisorSum[n,form] represents the sum ofform[i] for alli that dividen.DivisorSum[n,form,cond] includes only those divisors for whichcond[i] givesTrue.

This gives a list of sums for the divisors of five positive integers:

This imposes the condition that the value of each divisori must be less than 6:

TheJacobi symbolJacobiSymbol[n,m] reduces to theLegendre 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.

Theextended GCDExtendedGCD[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 first number in the list is the GCD of 105 and 196:

The second pair of numbers satisfies:

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

TheCarmichael function orleast universal exponent gives the smallest integer such that for all integers relatively prime to.

Continued fractions.

This generates the first 10 terms in the continued fraction representation for:

This reconstructs the number represented by the list of continued fraction terms:

The result is close to:

This gives directly a rational approximation 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.

Complete representations for numbers.

The continued fraction representation of starts with the term 8, then involves a sequence of terms that repeat forever:

This reconstructs from its continued fraction representation:

This number is a quadratic irrational:

This shows the recurring sequence of decimal digits in 3/7:

FromDigits reconstructs the original number:

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.

Continued fraction convergents.

This gives a list of rational approximations of 101/9801, derived from its continued fraction expansion:

This lists only the first 10 convergents:

This lists successive rational approximations to, until the numerical precision is exhausted:

With an exact irrational number, you have to explicitly ask for a certain number of terms:

Functions for integer lattices.

The lattice reduction functionLatticeReduce[{v₁,v₂,…}] is used in several kinds of modern algorithms. The basic idea is to think of the vectors of integers as defining a mathematicallattice. 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". WhatLatticeReduce does is to find a reduced set of basis vectors for the lattice, with certain special properties.

Three unit vectors along the three coordinate axes already form a reduced basis:

This gives the reduced basis for a lattice in four‐dimensional space specified by three vectors:

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 toRowReduce, 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.

In this case, the original matrix is recovered because it was in row echelon form:

This satisfies the required identities:

Here the second matrix has some pivots larger than 1, and nonzero entries over pivots:

Digit count function.

Here are the digits in the base 2 representation of the number 77:

This directly computes the number of ones in the base 2 representation:

The plot of the digit count function is self‐similar:

Bitwise operations.

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 then_i have ones.BitOr[n₁,n₂,…] yields the integer with ones at positions where any of then_i have ones.BitXor[n₁,n₂] yields the integer with ones at positions wheren₁ orn₂ but not both have ones.BitXor[n₁,n₂,…] has ones where an odd number of then_i have ones.

This finds the bitwise AND of the numbers 23 and 29 entered in base 2:

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 allowsBitNot[n] to be equivalent simply to.

Testing for a squared factor.

SquareFreeQ[n] checks to see ifn has a square prime factor. This is done by computingMoebiusMu[n] and seeing if the result is zero; if it is, thenn is not squarefree, otherwise it is. ComputingMoebiusMu[n] involves finding the smallest prime factorq ofn. Ifn has a small prime factor (less than or equal to), this is very fast. Otherwise,FactorInteger is used to findq.

This product of primes contains no squared factors:

The square number 4 divides 60:

SquareFreeQ can handle large integers:

Finding prime numbers.

NextPrime[n] finds the smallest primep such thatp>n. The algorithm does a direct search usingPrimeQ on the odd numbers greater thann.

This gives the next prime after 10:

Even for large numbers, the next prime can be computed rather quickly:

This gives the largest prime less than 34:

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

Here is a random prime between 10 and 100:

Testing for involving prime powers.

The algorithm forPrimePowerQ involves first computing the least prime factorp ofn and then attempting division byp until either 1 is obtained, in which casen is a prime power, or until division is no longer possible, in which casen is not a prime power.

Here is a number that is a power of a single prime:

Over theGaussianIntegers this is a prime power:

Solving simultaneous congruences.

The Chinese remainder theorem states that a certain class of simultaneous congruences always has a solution.ChineseRemainder[list₁,list₂] finds the smallest non‐negative integerr such thatMod[r,list₂] islist₁. The solution is unique modulo the least common multiple of the elements oflist₂.

This means that,, and:

This confirms the result:

Longer lists are still quite fast:

Computing primitive roots.

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

Here is a primitive root of 5:

This confirms that it does generate the group:

Here is a primitive root of a prime power:

Here is a primitive root of twice a prime power:

If the argument is composite and not a prime power or twice a prime power, the function does not evaluate:

Representing an integer as a sum of squares or other powers.

Here are the representations of 101 as a sum of 3 squares:

Combinatorial Functions

Combinatorial functions.

Thefactorial 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".

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

ThesubfactorialSubfactorial[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.

Themultinomial coefficientMultinomial[n₁,n₂,…], denoted, gives the number of ways of partitioning distinct objects into sets of sizes (with).

The Wolfram Language gives the exact integer result for the factorial of an integer:

For non‐integers, the Wolfram Language evaluates factorials using the gamma function:

The Wolfram Language can give symbolic results for some binomial coefficients:

This gives the number of ways of partitioning objects into sets containing 6 and 5 objects:

The result is the same as:

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

TheFibonacci polynomialsFibonacci[n,x] appear as the coefficients of in the expansion of.

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

TheBernoulli polynomialsBernoulliB[n,x] satisfy the generating function relation. TheBernoulli numbersBernoulliB[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 theGenocchi 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 usingBernoulliB[n], and then applyingN.

TheEuler polynomialsEulerE[n,x] have generating function, and theEuler numbersEulerE[n] are given by.

TheNörlund polynomialsNorlundB[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. Thegeneralized Bernoulli polynomialsNorlundB[n,a,x] satisfy the generating function relation.

This gives the second Bernoulli polynomial:

You can also get Bernoulli polynomials by explicitly computing the power series for the generating function:

BernoulliB[n] gives exact rational‐number results for Bernoulli numbers:

Stirling numbers show up in many combinatorial enumeration problems. ForStirling numbers of the first kindStirlingS1[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 kindStirlingS2[n,m], sometimes denoted, give the number of ways of partitioning a set of elements into non‐empty subsets. They satisfy the relation.

TheBell numbersBellB[n] give the total number of ways that a set of elements can be partitioned into non‐empty subsets. TheBell polynomialsBellB[n,x] satisfy the generating function relation.

Thepartition functionPartitionsP[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 lengthPartitionsP[n].

This gives a table of Stirling numbers of the first kind:

The Stirling numbers appear as coefficients in this product:

Here are the partitions of 4:

The number of partitions is given byPartitionsP[4]:

This gives the number of partitions of 100, with and without the constraint that the terms should be distinct:

The partition function increases asymptotically like. Note that you cannot simply usePlot to generate a plot of a function likePartitionsP because the function can only be evaluated with integer arguments:

Most of the functions here allow you tocount various kinds of combinatorial objects. Functions likeIntegerPartitions andPermutations allow you instead togenerate lists of various combinations of elements.

Thesignature functionSignature[{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 orepsilon symbol.

Rotational coupling coefficients.

Clebsch–Gordan coefficients and‐j symbols arise in the study of angular momenta in quantum mechanics, and in other applications of the rotation group. TheClebsch–Gordan coefficientsClebschGordan[{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.

The3‐j symbols orWigner coefficientsThreeJSymbol[{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.

The6‐j symbolsSixJSymbol[{j₁,j₂,j₃},{j₄,j₅,j₆}] give the couplings of three quantum mechanical angular momentum states. TheRacah coefficients are related by a phase to the 6‐j symbols.

You can give symbolic parameters in 3‐j symbols:

Elementary Transcendental Functions

Elementary transcendental functions.

The Wolfram Language gives exact results for logarithms whenever it can. Here is:

You can find the numerical values of mathematical functions to any precision:

This gives a complex number result:

The Wolfram Language can evaluate logarithms with complex arguments:

The arguments of trigonometric functions are always given in radians:

You can convert from degrees by explicitly multiplying by the constantDegree:

Here is a plot of the hyperbolic tangent function. It has a characteristic "sigmoidal" form:

Thehaversine functionHaversine[z] is defined by. Theinverse haversine functionInverseHaversine[z] is defined by. TheGudermannian functionGudermannian[z] is defined as. Theinverse Gudermannian functionInverseGudermannian[z] is defined by. The Gudermannian satisfies such relations as. Thesinc functionSinc[z] is the Fourier transform of a square signal.

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

Functions That Do Not Have Unique Values

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 functionSqrt[x] does.

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

gives, not:

Squaring and taking the square root does not necessarily give you the number 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.

This gives, not:

The branch cut inSqrt along the negative real axis means that values ofSqrt[z] with just above and below the axis are very different:

Their squares are nevertheless close:

The discontinuity along the negative real axis is quite clear in this three‐dimensional picture of the imaginary part of the square root function:

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

This takes the tenth power of a complex number. The result is unique:

There are 10 possible tenth roots. The Wolfram Language chooses one of them. In this case it is not the number whose tenth power you took:

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.

Some branch‐cut discontinuities in the complex plane.

ArcSin is a multiple‐valued function, so there is no guarantee that it always gives the "inverse" ofSin:

Values ofArcSin[z] on opposite sides of the branch cut can be very different:

A three‐dimensional picture, showing the two branch cuts for the function:

Mathematical Constants

Mathematical constants.

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

Khinchin's constantKhinchin (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 constantGlaisher (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.

Mathematical constants can be evaluated to arbitrary precision:

Exact computations can also be done with them:

Orthogonal Polynomials

Orthogonal polynomials.

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

Theassociated Legendre polynomialsLegendreP[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.

Thespherical harmonicsSphericalHarmonicY[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.

This gives the algebraic form of the Legendre polynomial:

The integral gives zero by virtue of the orthogonality of the Legendre polynomials:

Integrating the square of a single Legendre polynomial gives a nonzero result:

High‐degree Legendre polynomials oscillate rapidly:

The associated Legendre "polynomials" involve fractional powers:

"Special Functions" discusses the generalization of Legendre polynomials to Legendre functions, which can have noninteger degrees:

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

TheChebyshev polynomials of the second kindChebyshevU[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 polynomialsHermiteH[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 theparabolic cylinder functions orWeber functions by.

This gives the density for an excited state of a quantum‐mechanical harmonic oscillator. The average of the wiggles is roughly the classical physics result:

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

You can get formulas for generalized Laguerre polynomials with arbitrary values of:

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

Jacobi polynomialsJacobiP[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.

Special Functions

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.

The Wolfram System gives exact results for some values of special functions:

No exact result is known here:

A numerical result, to arbitrary precision, can nevertheless be found:

You can give complex arguments to special functions:

Special functions automatically get applied to each element in a list:

The Wolfram System knows analytic properties of special functions, such as derivatives:

You can useFindRoot to find roots of special functions:

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.

Gamma and related functions.

TheEuler gamma functionGamma[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 asLog[Gamma[z]]. For complex arguments, however, this form yields spurious discontinuities. The Wolfram System therefore includes the separate functionLogGamma[z], which yields thelogarithm of the gamma function with a single branch cut along the negative real axis.

TheEuler beta functionBeta[a,b] is.

ThePochhammer symbol orrising factorialPochhammer[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.

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

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

Theincomplete beta functionBeta[z,a,b] is given by. Notice that in the incomplete beta function, the parameter is anupper limit of integration, and appears as thefirst argument of the function. In the incomplete gamma function, on the other hand, is alower limit of integration, and appears as thesecond 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 computeregularized forms in which these functions are divided by complete beta and gamma functions. The Wolfram System includes theregularized incomplete beta functionBetaRegularized[z,a,b] defined for most arguments by, but taking into account singular cases. The Wolfram System also includes theregularized incomplete gamma functionGammaRegularized[a,z] defined by, with singular cases taken into account.

The incomplete beta and gamma functions, and their inverses, are common in statistics. Theinverse beta functionInverseBetaRegularized[s,a,b] is the solution for in. Theinverse gamma functionInverseGammaRegularized[a,s] is similarly the solution for in.

Derivatives of the gamma function often appear in summing rational series. Thedigamma functionPolyGamma[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.

Thepolygamma functionsPolyGamma[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 theGamma function and is defined by its functional identityBarnesG[z+1]=Gamma[z]BarnesG[z], where the third derivative of the logarithm ofBarnesG is positive for positivez.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 thatExp[LogBarnesG[z]]=BarnesG[z].

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

Many exact results for gamma and polygamma functions are built into the Wolfram System:

Here is a contour plot of the gamma function in the complex plane:

Zeta and related functions.

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

TheRiemann zeta functionZeta[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 twoRiemann–Siegel functionsRiemannSiegelZ[t] andRiemannSiegelTheta[t] according to and (for real). Note that the Riemann–Siegel functions are both real as long as is real.

TheStieltjes constantsStieltjesGamma[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.

Thegeneralized Riemann zeta functionZeta[s,a] is implemented as, where any term with is excluded.

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

The Ramanujan Dirichlet L-functionRamanujanTauL[s] is defined byL(s) (for), with coefficientsRamanujanTau[n]. In analogy with theRiemann zeta function, it is again convenient to define the functionsRamanujanTauZ[t] andRamanujanTauTheta[t].

Here is the numerical approximation for:

Here is a three-dimensional picture of the real part of a Dirichlet L-function:

The Wolfram System gives exact results for:

Here is a three‐dimensional picture of the Riemann zeta function in the complex plane:

This is a plot of the absolute value of the Riemann zeta function on the critical line. You can see the first few zeros of the zeta function:

This is a plot of the absolute value of the Ramanujan L function on its critical line:

Thepolylogarithm functionsPolyLog[n,z] are given by. The polylogarithm function is sometimes known asJonquière's function. ThedilogarithmPolyLog[2,z] satisfies. Sometimes is known asSpence's integral. TheNielsen generalized polylogarithm functions orhyperlogarithmsPolyLog[n,p,z] are given by. Polylogarithm functions appear in Feynman diagram integrals in elementary particle physics, as well as in algebraic K‐theory.

TheLerch transcendentLerchPhi[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, theCatalan beta function can be obtained as.

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

The Lerch transcendent can also be used to evaluateDirichlet L‐series that appear in number theory. The basicL‐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.

TheHurwitz–Lerch transcendentHurwitzLerchPhi[z,s,a] generalizesHurwitzZeta[s,a] and is defined by.

Zeros of the zeta function.

ZetaZero[1] represents the first nontrivial zero of:

This gives its numerical value:

This gives the first zero with height greater than 15:

Exponential integral and related functions.

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

Theexponential integral functionExpIntegralE[n,z] is defined by.

The secondexponential integral functionExpIntegralEi[z] is defined by (for), where the principal value of the integral is taken.

Thelogarithmic integral functionLogIntegral[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.

Thesine and cosine integral functionsSinIntegral[z] andCosIntegral[z] are defined by and. Thehyperbolic sine and cosine integral functionsSinhIntegral[z] andCoshIntegral[z] are defined by and.

Error function and related functions.

Theerror functionErf[z] is the integral of the Gaussian distribution, given by. Thecomplementary error functionErfc[z] is given simply by. Theimaginary error functionErfi[z] is given by. The generalized error functionErf[z₀,z₁] is defined by the integral. The error function is central to many calculations in statistics.

Theinverse error functionInverseErf[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 theFresnel integralsFresnelC[z] defined by andFresnelS[z] defined by. Fresnel integrals occur in diffraction theory.

Bessel and related functions.

TheBessel functionsBesselJ[n,z] andBesselY[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 theBessel function of the first kind, or simplythe Bessel function. is referred to as theBessel function of the second kind, theWeber function, or theNeumann function (denoted).

TheHankel functions (orBessel functions of the third kind)HankelH1[n,z] andHankelH2[n,z] give an alternative pair of solutions to the Bessel differential equation, related according to.

Thespherical Bessel functionsSphericalBesselJ[n,z] andSphericalBesselY[n,z], as well as thespherical Hankel functionsSphericalHankelH1[n,z] andSphericalHankelH2[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 usingFunctionExpand.