List

Functions

SpecialFunctions.erf — Function.
erf(x)

Compute the error function of $x$, defined by

\[\operatorname{erf}(x) = \frac{2}{\pi} \int_0^x \exp(-t^2) \; \mathrm{d}t \quad \text{for} \quad x \in \mathbb{C} \, .\]

External links: DLMF, Wikipedia.

See also: erfc(x), erfcx(x), erfi(x), dawson(x), erfinv(x), erfcinv(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
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SpecialFunctions.erfc — Function.
erfc(x)

Compute the complementary error function of $x$, defined by

\[\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\pi} \int_x^\infty \exp(-t^2) \; \mathrm{d}t \quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of 1-erf(x) for large $x$.

External links: DLMF, Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: C library for multiple-precision floating-point MPFR
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SpecialFunctions.erfcx — Function.
erfcx(x)

Compute the scaled complementary error function of $x$, defined by

\[\operatorname{erfcx}(x) = e^{x^2} \operatorname{erfc}(x) \quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of $e^{x^2} \operatorname{erfc}(x)$ for large $x$. Note also that $\operatorname{erfcx}(-ix)$ computes the Faddeeva function w(x).

External links: DLMF, Wikipedia.

See also: erfc(x).

Implementation by

  • Float32/Float64: C standard math library libm.
  • BigFloat: MPFR has an open TODO item for this function until then, we use DLMF 7.12.1 for the tail.
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SpecialFunctions.erfi — Function.
erfi(x)

Compute the imaginary error function of $x$, defined by

\[\operatorname{erfi}(x) = -i \operatorname{erf}(ix) \quad \text{for} \quad x \in \mathbb{C} \, .\]

External links: Wikipedia.

See also: erf(x).

Implementation by

  • Float32/Float64: C standard math library libm.
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SpecialFunctions.dawson — Function.
dawson(x)

Compute the Dawson function (scaled imaginary error function) of $x$, defined by

\[\operatorname{dawson}(x) = \frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x) \quad \text{for} \quad x \in \mathbb{C} \, .\]

This is the accurate version of $\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)$ for large $x$.

External links: DLMF, Wikipedia.

See also: erfi(x).

Implementation by

  • Float32/Float64: C standard math library libm.
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SpecialFunctions.erfinv — Function.
erfinv(x)

Compute the inverse error function of a real $x$, that is

\[\operatorname{erfinv}(x) = \operatorname{erf}^{-1}(x) \quad \text{for} \quad x \in \mathbb{R} \, .\]

External links: Wikipedia.

See also: erf(x).

Implementation

Using the rational approximants tabulated in:

J. M. Blair, C. A. Edwards, and J. H. Johnson, "Rational Chebyshev approximations for the inverse of the error function", Math. Comp. 30, pp. 827–830 (1976). http://dx.doi.org/10.1090/S0025-5718-1976-0421040-7, http://www.jstor.org/stable/2005402

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SpecialFunctions.erfcinv — Function.
erfcinv(x)

Compute the inverse error complementary function of a real $x$, that is

\[\operatorname{erfcinv}(x) = \operatorname{erfc}^{-1}(x) \quad \text{for} \quad x \in \mathbb{R} \, .\]

External links: Wikipedia.

See also: erfc(x).

Implementation

Using the rational approximants tabulated in:

J. M. Blair, C. A. Edwards, and J. H. Johnson, "Rational Chebyshev approximations for the inverse of the error function", Math. Comp. 30, pp. 827–830 (1976). http://dx.doi.org/10.1090/S0025-5718-1976-0421040-7, http://www.jstor.org/stable/2005402

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SpecialFunctions.sinint — Function.
sinint(x)

Compute the sine integral function of $x$, defined by

\[\operatorname{Si}(x) := \int_0^x \frac{\sin t}{t} \, \mathrm{d}t \quad \text{for} \quad x \in \mathbb{R} \,.\]

External links: DLMF, Wikipedia.

See also: cosint(x).

Implementation

Using the rational approximants tabulated in:

A.J. MacLeod, "Rational approximations, software and test methods for sine and cosine integrals", Numer. Algor. 12, pp. 259–272 (1996). http://dx.doi.org/10.1007/BF02142806, https://link.springer.com/article/10.1007/BF02142806.

Note: the second zero of $Ci(x)$ has a typo that is fixed: $r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402$ in the article, but is in fact: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

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SpecialFunctions.cosint — Function.
cosint(x)

Compute the cosine integral function of $x$, defined by

\[\operatorname{Ci}(x) := \gamma + \log x + \int_0^x \frac{\cos (t) - 1}{t} \, \mathrm{d}t \quad \text{for} \quad x > 0 \,,\]

where $\gamma$ is the Euler-Mascheroni constant.

External links: DLMF, Wikipedia.

See also: sinint(x).

Implementation

Using the rational approximants tabulated in:

A.J. MacLeod, "Rational approximations, software and test methods for sine and cosine integrals", Numer. Algor. 12, pp. 259–272 (1996). http://dx.doi.org/10.1007/BF02142806, https://link.springer.com/article/10.1007/BF02142806.

Note: the second zero of $Ci(x)$ has a typo that is fixed: $r_1 = 3.38418 0422\mathbf{8} 51186 42639 78511 46402$ in the article, but is in fact: $r_1 = 3.38418 0422\mathbf{5} 51186 42639 78511 46402$.

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SpecialFunctions.digamma — Function.
digamma(x)

Compute the digamma function of x (the logarithmic derivative of gamma(x)).

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SpecialFunctions.invdigamma — Function.
invdigamma(x)

Compute the inverse digamma function of x.

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SpecialFunctions.trigamma — Function.
trigamma(x)

Compute the trigamma function of x (the logarithmic second derivative of gamma(x)).

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SpecialFunctions.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))

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SpecialFunctions.airyai — Function.
airyai(x)

Airy function of the first kind $\operatorname{Ai}(x)$.

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SpecialFunctions.airyaiprime — Function.
airyaiprime(x)

Derivative of the Airy function of the first kind $\operatorname{Ai}'(x)$.

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SpecialFunctions.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.

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SpecialFunctions.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.

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SpecialFunctions.airybi — Function.
airybi(x)

Airy function of the second kind $\operatorname{Bi}(x)$.

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SpecialFunctions.airybiprime — Function.
airybiprime(x)

Derivative of the Airy function of the second kind $\operatorname{Bi}'(x)$.

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SpecialFunctions.airybix — Function.
airybix(x)

Scaled Airy function of the second kind $\operatorname{Bi}(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.

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SpecialFunctions.airybiprimex — Function.
airybiprimex(x)

Scaled derivative of the Airy function of the second kind $\operatorname{Bi}'(x) e^{- \left| \operatorname{Re} \left( \frac{2}{3} x \sqrt{x} \right) \right|}$.

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SpecialFunctions.besselj0 — Function.
besselj0(x)

Bessel function of the first kind of order 0, $J_0(x)$.

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SpecialFunctions.besselj1 — Function.
besselj1(x)

Bessel function of the first kind of order 1, $J_1(x)$.

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SpecialFunctions.besselj — Function.
besselj(nu, x)

Bessel function of the first kind of order nu, $J_\nu(x)$.

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SpecialFunctions.besseljx — Function.
besseljx(nu, x)

Scaled Bessel function of the first kind of order nu, $J_\nu(x) e^{- | \operatorname{Im}(x) |}$.

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SpecialFunctions.bessely0 — Function.
bessely0(x)

Bessel function of the second kind of order 0, $Y_0(x)$.

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SpecialFunctions.bessely1 — Function.
bessely1(x)

Bessel function of the second kind of order 1, $Y_1(x)$.

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SpecialFunctions.bessely — Function.
bessely(nu, x)

Bessel function of the second kind of order nu, $Y_\nu(x)$.

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SpecialFunctions.besselyx — Function.
besselyx(nu, x)

Scaled Bessel function of the second kind of order nu, $Y_\nu(x) e^{- | \operatorname{Im}(x) |}$.

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SpecialFunctions.hankelh1 — Function.
hankelh1(nu, x)

Bessel function of the third kind of order nu, $H^{(1)}_\nu(x)$.

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SpecialFunctions.hankelh1x — Function.
hankelh1x(nu, x)

Scaled Bessel function of the third kind of order nu, $H^{(1)}_\nu(x) e^{-x i}$.

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SpecialFunctions.hankelh2 — Function.
hankelh2(nu, x)

Bessel function of the third kind of order nu, $H^{(2)}_\nu(x)$.

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SpecialFunctions.hankelh2x — Function.
hankelh2x(nu, x)

Scaled Bessel function of the third kind of order nu, $H^{(2)}_\nu(x) e^{x i}$.

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SpecialFunctions.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.)

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SpecialFunctions.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.

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SpecialFunctions.besseli — Function.
besseli(nu, x)

Modified Bessel function of the first kind of order nu, $I_\nu(x)$.

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SpecialFunctions.besselix — Function.
besselix(nu, x)

Scaled modified Bessel function of the first kind of order nu, $I_\nu(x) e^{- | \operatorname{Re}(x) |}$.

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SpecialFunctions.besselk — Function.
besselk(nu, x)

Modified Bessel function of the second kind of order nu, $K_\nu(x)$.

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SpecialFunctions.besselkx — Function.
besselkx(nu, x)

Scaled modified Bessel function of the second kind of order nu, $K_\nu(x) e^x$.

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SpecialFunctions.ellipk — Function.
ellipk(m)

Computes Complete Elliptic Integral of 1st kind $K(m)$ for parameter $m$ given by

\[\operatorname{ellipk}(m) = K(m) = \int_0^{ \frac{\pi}{2} } \frac{1}{\sqrt{1 - m \sin^2 \theta}} \, \mathrm{d}\theta \quad \text{for} \quad m \in \left( -\infty, 1 \right] \, .\]

External links: DLMF, Wikipedia.

See also: ellipe(m).

Arguments

  • m: parameter $m$ is in relation to elliptic modulus $k$ by $k^2=m$ and modular angle $\alpha$ by $k=\sin \alpha$

Implementation

Using piecewise approximation polynomial as given in

'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions', Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr, DOI 10.1007/s10569-009-9228-z, https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf

For $m<0$`, followed

Fukushima, Toshio. (2014). 'Precise, compact, and fast computation of complete elliptic integrals by piecewise minimax rational function approximation'. Journal of Computational and Applied Mathematics. 282. DOI 10.13140/2.1.1946.6245., https://www.researchgate.net/publication/267330394

Also suggested in this paper that we should consider domain only from $(-\infty,1]$.

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SpecialFunctions.ellipe — Function.
ellipe(m)

Computes Complete Elliptic Integral of 2nd kind $E(m)$ for parameter $m$ given by

\[\operatorname{ellipe}(m) = E(m) = \int_0^{ \frac{\pi}{2} } \sqrt{1 - m \sin^2 \theta} \, \mathrm{d}\theta \quad \text{for} \quad m \in \left( -\infty, 1 \right] \, .\]

External links: DLMF, Wikipedia.

See also: ellipk(m).

Arguments

  • m: parameter $m$ is in relation to elliptic modulus $k$ by $k^2=m$ and modular angle $\alpha$ by $k=\sin \alpha$

Implementation

Using piecewise approximation polynomial as given in

'Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions', Fukushima, Toshio. (2014). F09-FastEI. Celest Mech Dyn Astr, DOI 10.1007/s10569-009-9228-z, https://pdfs.semanticscholar.org/8112/c1f56e833476b61fc54d41e194c962fbe647.pdf

For $m<0$`, followed

Fukushima, Toshio. (2014). 'Precise, compact, and fast computation of complete elliptic integrals by piecewise minimax rational function approximation'. Journal of Computational and Applied Mathematics. 282. DOI 10.13140/2.1.1946.6245., https://www.researchgate.net/publication/267330394

Also suggested in this paper that we should consider domain only from $(-\infty,1]$.

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SpecialFunctions.eta — Function.
eta(x)

Dirichlet eta function $\eta(s) = \sum^\infty_{n=1}(-1)^{n-1}/n^{s}$.

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SpecialFunctions.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 $\zeta(s)$.

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zeta(s)

Riemann zeta function $\zeta(s)$.

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SpecialFunctions.gamma — Function.
gamma(z)

Compute the gamma function for complex $z$, defined by

\[\Gamma(z) := \begin{cases} n! & \text{for} \quad z = n+1 \;, n = 0,1,2,\dots \\ \int_0^\infty t^{z-1} {\mathrm e}^{-t} \, {\mathrm d}t & \text{for} \quad \Re(z) > 0 \end{cases}\]

and by analytic continuation in the whole complex plane.

External links: DLMF, Wikipedia.

See also: loggamma(z).

Implementation by

  • Float: C standard math library libm.
  • Complex: by exp(loggamma(z)).
  • BigFloat: C library for multiple-precision floating-point MPFR
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SpecialFunctions.gamma_inc — Function.
gamma_inc(a,x,IND)

DLMF: https://dlmf.nist.gov/8.2#E4 , https://dlmf.nist.gov/8.2#E5 Wikipedia: https://en.wikipedia.org/wiki/Incompletegammafunction IND –> Accuracy desired ; IND=0 means 14 significant digits accuracy , IND=1 means 6 significant digit and IND=2 means only 3 digit accuracy suffices. gammainc(a,x) or P(a,x) is the Incomplete gamma function ratio given by : ``1/\Gamma (a) \int{0}^{x} e^{-t}t^{a-1} dt$gamma_q(a,x) or Q(a,x) is the Incomplete gamma function ratio given by : 1 - P(a,x) ->$1/\Gamma (a) \int{x}^{\infty} e^{-t}t^{a-1} dt`` Returns a tuple (gammainc, gammaq) where gammainc + gamma_q = 1.0

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SpecialFunctions.gamma_inc_inv — Function.
gamma_inc_inv(a,p,q)

DLMF: https://dlmf.nist.gov/8.2#E4 , https://dlmf.nist.gov/8.2#E5 Wiki: https://en.wikipedia.org/wiki/Incompletegammafunction

gammainc(a,x) or (P(a,x),Q(a,x)) is the Incomplete gamma function ratio given by : ``1/\Gamma (a) \int{0}^{x} e^{-t}t^{a-1} dt`gamma_inc_inv(a,p,q) inverts the gamma_inc function, by computingxgivena,p,q` in P(a,x)=p and Q(a,x)=q.

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SpecialFunctions.beta_inc — Function.
beta_inc(a,b,x,y)

Computes Incomplete Beta Function Ratios given by:

\[I_{x}(a,b) = G(a,b) \int_{0}^{x} t^{a-1}(1-t)^{b-1} dt,\]

and $I_{y}(a,b) = 1.0 - I_{x}(a,b)$. given $B(a,b) = 1/G(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$ and $x+y = 1$.

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SpecialFunctions.loggamma — Function.
loggamma(x)

Computes the logarithm of gamma for given x. If x is a Real, then it throws a DomainError if gamma(x) is negative.

See also logabsgamma.

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SpecialFunctions.logabsgamma — Function.
logabsgamma(x)

Compute the logarithm of absolute value of gamma for Real xand returns a tuple (log(abs(gamma(x))), sign(gamma(x))).

See also loggamma.

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SpecialFunctions.logfactorial — Function.
logfactorial(x)

Compute the logarithmic factorial of a nonnegative integer x. Equivalent to loggamma of x + 1, but loggamma extends this function to non-integer x.

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SpecialFunctions.beta — Function.
beta(x, y)

Euler integral of the first kind $\operatorname{B}(x,y) = \Gamma(x)\Gamma(y)/\Gamma(x+y)$.

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SpecialFunctions.logbeta — Function.
logbeta(x, y)

Natural logarithm of the beta function $\log(|\operatorname{B}(x,y)|)$.

See also logabsbeta.

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SpecialFunctions.logabsbeta — Function.
logabsbeta(x, y)

Compute the natural logarithm of the absolute value of the beta function, returning a tuple (log(abs(beta(x,y))), sign(beta(x,y)))

See also logbeta.

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SpecialFunctions.logabsbinomial — Function.
logabsbinomial(n, k)

Accurate natural logarithm of the absolute value of the binomial coefficient binomial(n, k) for large n and k near n/2.

Returns a tuple (log(abs(binomial(n,k))), sign(binomial(n,k))).

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