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), 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 $x$ |