Functions
Base.erf — Function.erf(x)Compute the error function of x, defined by $\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt$ for arbitrary complex x.
Base.erfc — Function.erfc(x)Compute the complementary error function of x, defined by $1 - \operatorname{erf}(x)$.
Base.erfcx — Function.erfcx(x)Compute the scaled complementary error function of x, defined by $e^{x^2} \operatorname{erfc}(x)$. Note also that $\operatorname{erfcx}(-ix)$ computes the Faddeeva function $w(x)$.
Base.erfi — Function.erfi(x)Compute the imaginary error function of x, defined by $-i \operatorname{erf}(ix)$.
Base.dawson — Function.dawson(x)Compute the Dawson function (scaled imaginary error function) of x, defined by $\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)$.
Base.erfinv — Function.erfinv(x)Compute the inverse error function of a real x, defined by $\operatorname{erf}(\operatorname{erfinv}(x)) = x$.
Base.erfcinv — Function.erfcinv(x)Compute the inverse error complementary function of a real x, defined by $\operatorname{erfc}(\operatorname{erfcinv}(x)) = x$.
Base.digamma — Function.digamma(x)Compute the digamma function of x (the logarithmic derivative of gamma(x)).
Base.invdigamma — Function.invdigamma(x)Compute the inverse digamma function of x.
Base.trigamma — Function.trigamma(x)Compute the trigamma function of x (the logarithmic second derivative of gamma(x)).
Base.polygamma — Function.polygamma(m, x)Compute the polygamma function of order m of argument x (the (m+1)th derivative of the logarithm of gamma(x))
Base.airyai — Function.airyai(x)Airy function of the first kind $\operatorname{Ai}(x)$.
Base.airyaiprime — Function.airyaiprime(x)Derivative of the Airy function of the first kind $\operatorname{Ai}'(x)$.
Base.airyaix — Function.airyaix(x)Scaled Airy function of the first kind $\operatorname{Ai}(x) e^{\frac{2}{3} x \sqrt{x}}$. Throws DomainError for negative Real arguments.
Base.airyaiprimex — Function.airyaiprimex(x)Scaled derivative of the Airy function of the first kind $\operatorname{Ai}'(x) e^{\frac{2}{3} x \sqrt{x}}$. Throws DomainError for negative Real arguments.
Base.airybi — Function.airybi(x)Airy function of the second kind $\operatorname{Bi}(x)$.
Base.airybiprime — Function.airybiprime(x)Derivative of the Airy function of the second kind $\operatorname{Bi}'(x)$.
Base.airybix — Function.airybix(x)Scaled Airy function of the second kind $\operatorname{Bi}(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.
Base.airybiprimex — Function.airybiprimex(x)Scaled derivative of the Airy function of the second kind $\operatorname{Bi}'(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.
Base.besselj0 — Function.besselj0(x)Bessel function of the first kind of order 0, $J_0(x)$.
Base.besselj1 — Function.besselj1(x)Bessel function of the first kind of order 1, $J_1(x)$.
Base.besselj — Function.besselj(nu, x)Bessel function of the first kind of order nu, $J_\nu(x)$.
Base.besseljx — Function.besseljx(nu, x)Scaled Bessel function of the first kind of order nu, $J_\nu(x) e^{- | \operatorname{Im}(x) |}$.
Base.bessely0 — Function.bessely0(x)Bessel function of the second kind of order 0, $Y_0(x)$.
Base.bessely1 — Function.bessely1(x)Bessel function of the second kind of order 1, $Y_1(x)$.
Base.bessely — Function.bessely(nu, x)Bessel function of the second kind of order nu, $Y_\nu(x)$.
Base.besselyx — Function.besselyx(nu, x)Scaled Bessel function of the second kind of order nu, $Y_\nu(x) e^{- | \operatorname{Im}(x) |}$.
Base.hankelh1 — Function.hankelh1(nu, x)Bessel function of the third kind of order nu, $H^{(1)}_\nu(x)$.
Base.hankelh1x — Function.hankelh1x(nu, x)Scaled Bessel function of the third kind of order nu, $H^{(1)}_\nu(x) e^{-x i}$.
Base.hankelh2 — Function.hankelh2(nu, x)Bessel function of the third kind of order nu, $H^{(2)}_\nu(x)$.
Base.hankelh2x — Function.hankelh2x(nu, x)Scaled Bessel function of the third kind of order nu, $H^{(2)}_\nu(x) e^{x i}$.
Base.besselh — Function.besselh(nu, [k=1,] x)Bessel function of the third kind of order nu (the Hankel function). k is either 1 or 2, selecting hankelh1 or hankelh2, respectively. k defaults to 1 if it is omitted. (See also besselhx for an exponentially scaled variant.)
Base.besselhx — Function.besselhx(nu, [k=1,] z)Compute the scaled Hankel function $\exp(∓iz) H_ν^{(k)}(z)$, where $k$ is 1 or 2, $H_ν^{(k)}(z)$ is besselh(nu, k, z), and $∓$ is $-$ for $k=1$ and $+$ for $k=2$. k defaults to 1 if it is omitted.
The reason for this function is that $H_ν^{(k)}(z)$ is asymptotically proportional to $\exp(∓iz)/\sqrt{z}$ for large $|z|$, and so the besselh function is susceptible to overflow or underflow when z has a large imaginary part. The besselhx function cancels this exponential factor (analytically), so it avoids these problems.
Base.besseli — Function.besseli(nu, x)Modified Bessel function of the first kind of order nu, $I_\nu(x)$.
Base.besselix — Function.besselix(nu, x)Scaled modified Bessel function of the first kind of order nu, $I_\nu(x) e^{- | \operatorname{Re}(x) |}$.
Base.besselk — Function.besselk(nu, x)Modified Bessel function of the second kind of order nu, $K_\nu(x)$.
Base.besselkx — Function.besselkx(nu, x)Scaled modified Bessel function of the second kind of order nu, $K_\nu(x) e^x$.
Base.eta — Function.eta(x)Dirichlet eta function $\eta(s) = \sum^\infty_{n=1}(-1)^{n-1}/n^{s}$.
Base.zeta — Function.zeta(s, z)Generalized zeta function $\zeta(s, z)$, defined by the sum $\sum_{k=0}^\infty ((k+z)^2)^{-s/2}$, where any term with $k+z=0$ is excluded. For $\Re z > 0$, this definition is equivalent to the Hurwitz zeta function $\sum_{k=0}^\infty (k+z)^{-s}$. For $z=1$, it yields the Riemann zeta function $\zeta(s)$.
zeta(s)Riemann zeta function $\zeta(s)$.