SpecialFunctions パッケージに含まれる関数一覧

# SpecialFunctions パッケージに含まれる関数一覧

## Gamma Function

Function | Description
:-----|:-----
gamma(z) | gamma function $\Gamma(z)$
loggamma(x) | accurate log(gamma(x)) for large $x$
logabsgamma(x) | accurate log(abs(gamma(x))) for large $x$
logfactorial(x) | accurate log(factorial(x)) for large $x$; same as loggamma(x+1) for $x \gt 1$, zero otherwise
digamma(x) | digamma function (i.e. the derivative of loggamma at $x$)
invdigamma(x) | invdigamma function (i.e. inverse of digamma function at x using fixedpoint iteration algorithm)
trigamma(x) | trigamma function (i.e the logarithmic second derivative of gamma at $x$)
polygamma(m,x) | polygamma function (i.e the (m+1)-th derivative of the loggamma function at $x$)
gamma(a,z) | upper incomplete gamma function $\Gamma(a, z)$
loggamma(a,z) | accurate log(gamma(a,x)) for large arguments
gamma_inc(a,x,IND) | incomplete gamma function ratio P(a,x) and Q(a,x) (i.e evaluates P(a,x) and Q(a,x)for accuracy specified by IND and returns tuple (p,q))
beta_inc(a,b,x,y) | incomplete beta function ratio Ix(a,b) and Iy(a,b) (i.e evaluates Ix(a,b) and Iy(a,b) and returns tuple (p,q))
gamma_inc_inv(a,p,q) | inverse of incomplete gamma function ratio P(a,x) and Q(a,x) (i.e evaluates x given P(a,x)=p and Q(a,x)=q
beta(x,y) | beta function at $x$, $y$
logbeta(x,y)         | accurate log(beta(x,y)) for large $x$ or $y$
logabsbeta(x,y)    | accurate log(abs(beta(x,y))) for large $x$ or $y$
logabsbinomial(x,y)    | accurate log(abs(binomial(n,k))) for large $n$ and $k$ near n/2

## Exponential and Trigonometric Integrals

Function | Description
:-----|:-----
expint(ν, z)   | exponential integral $\text{E}_\nu(z)$
expinti(x)    | exponential integral $\text{E}_i(x)$
expintx(x)    | scaled exponential integral $e^z$ $\text{E}_\nu(z)$
sinint(x)    | sine integral $\text{S}_i(x)$
cosint(x)    | cosine integral $\text{C}_i(x)$

## Error Functions, Dawson’s and Fresnel Integrals

Function | Description
:-----|:-----
erf(x)    | error function at $x$
erf(x,y)    | accurate version of $\text{erf}(y) − \text{erf}(x)$
erfc(x)    | complementary error function, i.e. the accurate version of $1 − \text{erf} (x)$ for large $x$
erfcinv(x)    | inverse function to $\text{erfc}()$
erfcx(x)    | scaledcomplementaryerrorfunction,i.e.accurate $e^{x^2} \text{erfc}(x)$ for large $x$
logerfc(x)    | log of the complementary error function, i.e. accurate $\ln(\text{erfc}(x))$ for large $x$
logerfcx(x)    | log of the scaled complementary error function, i.e. accurate $\ln(\text{erfc}(x))$ for large negative $x$
erfi(x)    | imaginary error function defined as $−i\ \text{erf} (ix)$
erfinv(x)    | inverse function to $\text{erf}()$
dawson(x)    | scaled imaginary error function, a.k.a. Dawson function, i.e. accurate $\displaystyle \frac{\sqrt{\pi}}{2}e^{-x^2}\text{erfi}(x)$ for large $x$

## Airy and Related Functions

Function | Description
:-----|:-----
airyai(z)                | Airy Ai function at $z$
airyaiprime(z)            | derivative of the Airy Ai function at $z$
airybi(z)                | Airy Bi function at $z$
airybiprime(z)            | derivative of the Airy Bi function at $z$
airyaix(z), airyaiprimex(z),<br>airybix(z), airybiprimex(z)           | scaled Airy Ai function and kth derivatives at $z$

## Bessel Functions

Function | Description
:-----|:-----
besselj(nu,z)    | Bessel function of the first kind of order nu at $z$
besselj0(z)    | besselj(0,z)
besselj1(z)    | besselj(1,z)
besseljx(nu,z)    | scaled Bessel function of the first kind of order nu at $z$
sphericalbesselj(nu,z)    | Spherical Bessel function of the first kind of order nu at $z$
bessely(nu,z)    | Bessel function of the second kind of order nu at $z$
bessely0(z)    | bessely(0,z)
bessely1(z)    | bessely(1,z)
besselyx(nu,z)    | scaled Bessel function of the second kind of order nu at $z$
sphericalbessely(nu,z)    | Spherical Bessel function of the second kind of order $\nu$ at $z$
besselh(nu,k,z)    | Bessel function of the third kind (a.k.a. Hankel function) of order $\nu$ at $z$; $k$ must be either 1 or 2
hankelh1(nu,z)    | besselh(nu, 1, z)
hankelh1x(nu,z)    | scaled besselh(nu, 1, z)
hankelh2(nu,z)    | besselh(nu, 2, z)
hankelh2x(nu,z)         | scaled besselh(nu, 2, z)
besseli(nu,z)        | modified Bessel function of the first kind of order $\nu$ at $z$
besselix(nu,z)        | scaled modified Bessel function of the first kind of $\nu$ at $z$
besselk(nu,z)        | modified Bessel function of the second kind of order $\nu$ at $z$
besselkx(nu,z)        | scaled modified Bessel function of the second kind of $\nu$ at $z$
jinc(x)            | scaled Bessel function of the first kind divided by x. A.k.a. sombrero or besinc

## Elliptic Integrals

Function | Description
:-----|:-----
ellipk(m)    | complete elliptic integral of 1st kind $K(m)$
ellipe(m)     | complete elliptic integral of 2nd kind $E(m)$

## Zeta and Related Functions

Function | Description
:-----|:-----
eta(x) | Dirichlet eta function at $x$
zeta(x) | Riemann zeta function at $x$