## Special Functions

### AiryA(x)

The AiryA function returns the value of the Airy Ai (x ) function:

where K() is the modified Bessel function of the second kind.

The AiryAD function returns the value of the derivative of the Airy function.

### AiryB(x)

The AiryB function returns the value of the Airy Bi (x ) function:

where I() is the modified Bessel function of the first kind.

### AiryBD(x)

The AiryBD function returns the value of the derivative Bi' (x ) of the AiryB function.

### BesselI(n, z)

The BesselI function returns the modified Bessel function of the first kind, of order n and argument z. If z is real, a real value is returned. If z is real and negative, Besseli returns NaN unless n is an integer.

For complex z a complex value is returned, and there are no restrictions on z except for possible overflow.

### BesselJ(n, z)

The BesselJ function returns the Bessel function of the first kind, Jn (z), of ordern and argument z. If z is real, a real value is returned. If z is real and negative, BesselJ returns NaN unless n is an integer. For complex z a complex value is returned, and there are no restrictions on z except for possible overflow.

### BesselK(n, z)

The BesselK function returns the modified Bessel function of the second kind, Kn (z), of ordern and argument z. If z is real, a real value is returned. If z is real and negative, BesselK returns NaN.

### BesselY(n, z)

The BesselY function returns the Bessel function of the second kind, Yn (z ), of ordern and argument z. If z is real, a real value is returned. If z is real and negative, BesselY returns NaN.

### Beta(a, b)

The beta function returns for real or complex arguments

with Re(a), Re(b)>0.

### Betai

The betai function returns the regularized incomplete beta function

Here a,b>0, and 0<=x<=1.

### Binomial(n,k)

The binomial function returns the ratio:

where both n and k are positive integers, k<=n and ! denotes the factorial function.Binomialln(n,k)

Returns the natual log of the binomial coefficient for n, and k.

### Chebyshev(n,x)

The chebyshev function returns the Chebyshev polynomial of the first kind and of degree n. The Chebyshev polynomials satisfy the recurrence relation:

The orthogonality of the polynomial is expressed by the integral:

### ChebyshevU(n,x)

The chebyshevU function returns the Chebyshev polynomial of the second kind, degree n and argument x. The Chebyshev polynomial of the second kind satisfies the recurrence relation

U(n+1,x)=2xU(n,x)-U(n-1,x),

which is also the recurrence relation of the Chebyshev polynomials of the first kind. The first 10 polynomials of the second kind are:

U(0,x)=1

U(1,x)=2x

U(2,x)=4x^2-1

U(3,x)=8x^3-4x

U(4,x)=16x^4-12x^2+1

U(5,x)=32x^5-32x^3+6x

U(6,x)=64x^6-80x^4+24x-1

U(7,x)=128x^7-192x^5+80x^3-8x

U(8,x)=256x^8-448x^6+240x^4-40x^2+1

U(9,x)512x^9-1024x^7+672x^5-160x^3+10x

### Dawson(x)

The dawson function returns the value of the Dawson integral

### Digamma(z)

The digamma function returns the digamma, or psi function of z. This is the logarithmic derivative of the gamma function

In complex expressions, z is complex, and digamma(z) returns a complex value. Limited testing indicates that the accuracy is approximately 1 part in 1016, at least for moderately-sized values of x.

### Ei(x)

The ei function returns the value of the exponential integral Ei (x )

where P denotes the principal value of the integral.

### Erf(z)

The erf function returns the error function. For real input x the function is given by

In complex expressions the error function is defined by

where

is the confluent hypergeometric function of the first kind HyperG1F1.

### Erfc(z)

The erfc function returns the complementary error function of z (erfc(z) = 1 - erf(z)).

where

is the confluent hypergeometric function of the first kind HyperG1F1.

### expInt(n,x)

The expInt function returns the value of the exponential integral En (x )

where P is the principal value.

### Factorial(n)

The Factorial function returns n !, where n is assumed to be a positive integer. Note that while factorial is an integer-valued function, a double-precision number has 53 bits for the mantissa. This means that numbers over 252 will be accurate to about one part in about 2x1016. Values of n greater than 170 result in overflow and return the nonnumber Inf.

### FresnelCos(x)

The FresnelCos function returns the Fresnel cosine function C (x ).

### FresnelSin(x)

The FresnelSin function returns the Fresnel sine function S(x )

### Gamma(z)

The gammma function returns the value of the gamma function of z. If z is complex, it returns a complex result. Note that the return value for num close to negative integers is NaN, not +/-Inf.

### GammaInc(a,x)

The gammaInc function returns the value of the incomplete gamma function, defined by the integral

Note that gammaInc(a, x) = gamma(a) - gammaInc(a, x, 0). Defined for x > 0, a >= 0

### Gammln(z)

The gammln function returns the natural log of the gamma function of z, where z>0. If z is complex, it returns a complex result.

### Gammp(a,x)

The gammp function returns the regularized incomplete gamma function P(a,x), where a>0, x>=0. It is defined by

gammp(a,x)=gammaInc(a,x)/gamma(a).

### Gammq(a,x)

The gammq function returns the regularized incomplete gamma function 1-P(a,x), where a >0, x>=0. It is defined by

gammaInc(a,x)/gamma(a).

### Hermite(n,x)

The hermite function returns the Hermite polynomial of order n

The first few polynomials are:

H(0,x)=1

H(1,x)=2x

H(2,x)=4x^2-2

H(3,x)=8x^3-12x

### HermiteGauss(n,x)

The HermiteGauss function returns the normalized Hermite polynomial of order n :

Here the normalization was chosen such that

### HyperG0F1(b,z)

The hyperG0F1 function returns the confluent hypergeometric function

where

is the gamma function.

### HyperG1F1(a,b,z)

The hyperG1F1 function returns the confluent hypergeometric function

where the Pochhammer symbol is defined as:

### HyperG2F1(a,b,c,z)

The hyperG2F1 function returns the confluent hypergeometric function

where the Pochhammer symbol is defined as:

### HyperGPFQ(a,b,z)

The hyperGPFQ function returns the generalized hypergeometric function

where the Pochhammer symbol is defined as:

### InverseERF(x)

The inverseErf function returns the inverse of the error function

### InverseERFC

The inverseErfc function returns the inverse of the complementary error function.

### Laguerre

The laguerre function returns the Laguerre polynomial of degree n (positive integer) and argument x. The polynomials satisfy the recurrence relation:

with the initial conditions

### LegendreA

The legendreA function returns the associated Legendre polynomial

where n and m are integers such that 0<=m<=n and |x | <=1.

The first three polynomials are given by

### SphericalBessJ

The sphericalBessJ function returns the spherical Bessel function of the first kind and order n.

The first three functions are

### SphericalBessJD

The sphericalBessJD function returns the derivative of the spherical Bessel function of the first kind and order n.

### SphericalBessY(n,x)

The sphericalBessY function returns the spherical Bessel function of the second kind and order n.

The first few orders are given by

### SphericalBessYD(n,x)

The sphericalBessYD function returns the derivative of the spherical Bessel function of the second kind and order n.

### SphericalHarmonics(L,M,t,f)

The sphericalHarmonics function returns the complex-valued spherical harmonics

where

is the associated Legendre function.

### ZernikeR(n,m,r)

The ZernikeR function returns the Zernike radial polynomials of degree n that contains no power of r that is less than m. Here m is even or odd according to whether n is even or odd, and r is in the range [0,1]. Note that the full circle polynomials are complex. For any angle t (theta), they are given by: ZernikeR(n,m,r )*exp(it ).