Error function, and related functions

Exponentially scaled complementary error function, \(\mathrm{erfcx}(x)\)#

math53.erfcx(x)#

Returns the exponentially scaled complementary error function \(\displaystyle \mathrm{erfcx}(z) = \exp(z^2) \cdot \mathrm{erfc}(z) = w(iz)\). See also Ehrhardt [297] (3.3.6).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Erfce(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Erfce('0.51')
ereal('5.3518479027559984754E-1')

Difference of error functions, \(\mathrm{erfh}(x,h) = \mathrm{erf}(x+h)-\mathrm{erf}(x-h)\)#

math53.erfh(x, h)#

Returns the difference of error functions \(\displaystyle \mathrm{erfh}(x,h) = \mathrm{erf}(x+h)-\mathrm{erf}(x-h) = \mathrm{erfc}(x-h)-\mathrm{erfc}(x+h)\).

Difference of error functions, \(\mathrm{erf2}(x_1,x_2) = \mathrm{erf}(x_2)-\mathrm{erf}(x_1)\)#

math53.erf2(x1, x2)#

Returns the difference of error functions \(\displaystyle \mathrm{erf2}(x_1,x_2) = \mathrm{erf}(x_2)-\mathrm{erf}(x_1) = \mathrm{erfh}\left( \tfrac{1}{2}(x_2+x_1), \tfrac{1}{2}(x_2-x_1) \right)\).

Probability function \(Q(x) = \Phi(-x)\)#

math53.erfq(x)#

Returns the integral \(\displaystyle Q(x) = \Phi(-x) = \frac{1}{\sqrt 2\pi} \int_x^{\infty} \exp(-t^2)\, \mathrm{d}t = P(-x)\).

Inverse of the exponentially scaled complementary error function, \(\mathrm{erfcx}^{-1}(x)\)#

math53.erfcx_inv(x)#

Returns \(\mathrm{erfcx}^{-1}(x)\), the functional inverse of \(\mathrm{erfcx}\), satisfying \(\mathrm{erfcx}(\mathrm{erfcx}^{-1}(x)) = x\), for \(0 \le x \le \infty\).

Inverse of the imaginary error function, \(\mathrm{erfi}^{-1}(x)\)#

math53.erfi_inv(x)#

Returns \(\mathrm{erfi}^{-1}(x)\), the functional inverse of \(\mathrm{erfi}\), satisfying \(\mathrm{erfi}(\mathrm{erfi}^{-1}(x)) = x\).

Generalized Dawson integral, \(F(p, x)\)#

math53.dawson2(p, x)#

Returns the generalized Dawson integral \(\displaystyle F(p, x) = e^{-x^p} \int_0^x e^{t^p} \mathrm{d}t, p \ge 0, x \ge 0\). See also Ehrhardt [297] (3.3.2).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Dawson2(1.5, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Dawson2(1.5, '0.51')
ereal('5.3518479027559984754E-1')

Generalized error function, \(\mathrm{erfg}(p, x)\)#

math53.erfg(p, x)#

Returns the generalized error function \(\displaystyle \mathrm{erfg}(p, x) = \int_0^x e^{-t^p} \mathrm{d}t = \frac{1}{p}\gamma\left(\frac{1}{p}, x^p \right)\), where \(\gamma(\cdot)\) denotes the non-normalised lower incomplete gamma function. See also Ehrhardt [297] (3.3.4).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Erfg(1.5, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Erfg(1.5, '0.51')
ereal('5.3518479027559984754E-1')

Expint3, \(\mathrm{erfg}(3, x)\)#

math53.expint3(x)#

Returns \(\displaystyle \mathrm{expint3}(p, x) = \mathrm{erfg}(3, x) = \int_0^x e^{-t^3} \mathrm{d}t = \frac{1}{3}\gamma\left(\frac{1}{3}, x^3 \right)\), where \(\gamma(\cdot)\) denotes the non-normalised lower incomplete gamma function. See also Ehrhardt [297] (3.3.13).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Expint3(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Expint3('0.51')
ereal('5.3518479027559984754E-1')

Scaled repeated integrals of the complementary error function, \(i^n \mathrm{erfc}(x)\)#

math53.inerfc(n, x)#

Returns the scaled repeated integrals of the complementary error function, defined for \(n \ge -1\) using the awkward but standard notation

\[i^n \mathrm{erfc}(x) = \int_x^{\infty} \mathrm{erfc}(t) \, \mathrm{d}t = \frac{2}{\sqrt{\pi}} \int_x^{\infty} \frac{(t-x)^n}{n!} e^{-t^2} \, \mathrm{d}t, \quad (n = 0,1,2,...).\]

See also Ehrhardt [297] (3.3.7), NIST [438] (eq. 7.18.10), MathWorld [978].

These functions compute the scaled repeated integrals of complementary error function, defined for \(n \geq -1\) using the awkward but standard notation

\[i^n \text{erfc}(x) = \int_x^\infty i^{n-1} \text{erfc}(t)\mathrm{d}t = \frac{2}{\sqrt{\pi}} \int_x^\infty \frac{(t-x)^n}{n!} e^{-t^2} \mathrm{d}t, \quad (n=0,1,2,\ldots)\]

\[i^{-1} \text{erfc}(x) = \frac{2}{\sqrt{\pi}} e^{-x^2}, \quad i^{0} \text{erfc}(x) = \text{erfc}(x).\]

\[i^{n} \text{erfc}(x) = -\frac{z}{n} i^{n-1} \text{erfc}(x) + \frac{1}{2n} i^{n-2} \text{erfc}(x.\]

We also have

\[i^{n} \text{erfc}(x) = \frac{e^{-z^2}}{2^n \sqrt{\pi}} U \left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^2 \right)\]

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.InErfc(3, 0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.InErfc(3, '0.51')
ereal('5.3518479027559984754E-1')

An example with real input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; n = '4'; x = '5.0'
>>> \mathrm{d}x = dec.inerfc(n, x); mx = mpm.inerfc(n, x); gx = gmp.inerfc(n, x)
>>> fx = fpm.inerfc(n, x); ax = apm.inerfc(n, x)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
dec:  8.631401306269731883125812071651943647995E-6
mpm:  8.631401306269731883125812071651943647995e-6
gmp:  8.631401306269731883125812071651943647995E-06
fpm:  8.63140130626973E-06
apm:  8.631401306269731883125812071651943657349e-6 (1.945e-32%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; n = '4'; z = '5.0 + 3.0j'
>>> \mathrm{d}z = dec.inerfc(n, z); mz = mpm.inerfc(n, z); gz = gmp.inerfc(n, z)
>>> fz = fpm.inerfc(n, z); az = apm.inerfc(n, z)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: -3.8136067248438755019E-6               - 2.7044284620493310438E-6j
mpm: -3.8136067248438755019e-6               - 2.7044284620493310438e-6j
gmp: -3.8136067248438755019E-06              - 2.7044284620493310438E-06j
fpm: -3.81360672484388E-06                   - 2.70442846204933E-06j
apm: -3.8136067248438754005e-6 (-2.848e-13%) - 2.7044284620493310996e-6 (-3.025e-13%)j

Fresnel auxiliary function \(\mathrm{f}(x)\)#

math53.fresnel_f(x)#

Returns the Fresnel auxiliary function \(\displaystyle \mathrm{f}\left(x\right)=\left(\tfrac{1}{2}-S\left(x\right)\right)\cos\left(\tfrac{1}{2}\pi x^{2}\right)-\left(\tfrac{1}{2}-C\left(x\right)\right)\sin\left(\tfrac{1}{2}\pi x^{2}\right)\).

See also Wikipedia [1324], MathWorld [981], NIST [778], Ehrhardt [297] (3.3.15).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.FresnelF(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.FresnelF('0.51')
ereal('5.3518479027559984754E-1')

Fresnel auxiliary function \(\mathrm{g}(x)\)#

math53.fresnel_g(x)#

Returns the Fresnel auxiliary function \(\displaystyle \mathrm{g}\left(x\right)=\left(\tfrac{1}{2}-C\left(x\right)\right)\cos\left(\tfrac{1}{2}\pi x^{2}\right)+\left(\tfrac{1}{2}-S\left(x\right)\right)\sin\left(\tfrac{1}{2}\pi x^{2}\right)\).

See also Wikipedia [1324], MathWorld [981], NIST [778], Ehrhardt [297] (3.3.15).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.FresnelG(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.FresnelG('0.51')
ereal('5.3518479027559984754E-1')

Goodwin-Staton integral, \(G(x)\)#

math53.goodwin_staton(x)#

Returns the Goodwin-Staton integral \(\displaystyle G(x) = \int_0^{\infty} \frac{e^{-t^2}}{t+x} \, \mathrm{d}t = \sqrt{\pi} F(x) - \tfrac{1}{2} e^{-x^2} \mathrm{Ei}(x^2), \quad x \ne 0\).

See also Wikipedia [1325], NIST [437], Ehrhardt [297] (3.3.16).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.GoodwinStaton(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.GoodwinStaton('0.51')
ereal('5.3518479027559984754E-1')

Voigt function U#

math53.voigt_u(z)#

Returns the Voigt function U. See also Wikipedia [1338], NIST [440].

The Voigt functions[1] U, V, and H (sometimes called the line broadening function) are defined by

\[U(x,t)+iV(x,t)={\sqrt {\frac {\pi }{4t}}}e^{z^{2}}\operatorname {erfc} (z)={\sqrt {\frac {\pi }{4t}}}w(iz),\]

\[H(a,u)={\frac {U(u/a,1/4a^{2})}{a{\sqrt {\pi }}}},\]

where

\[z=(1-ix)/2{\sqrt {t}},\]

erfc is the complementary error function, and w(z) is the Faddeeva function.

An example in Python

>>> from xlcalcnet import ecplx
>>> ecplx.VoigtU(0.5)
ecplx('5.2359877559829887307E-1')
>>> ecplx.VoigtU('0.1')
ecplx('5.3518479027559984754E-1')

Voigt function V#

math53.voigt_v(x, t)#

Returns the Voigt function V. See also Wikipedia [1338], NIST [440].

An example in Python

>>> from xlcalcnet import ecplx
>>> ecplx.VoigtV(0.5)
ecplx('5.2359877559829887307E-1')
>>> ecplx.VoigtV('0.1')
ecplx('5.3518479027559984754E-1')

Voigt function H#

math53.voigt_h(x, t)#

Returns the Voigt function H. See also Wikipedia [1338], NIST [440].

An example in Python

>>> from xlcalcnet import ecplx
>>> ecplx.VoigtH(0.5)
ecplx('5.2359877559829887307E-1')
>>> ecplx.VoigtH('0.1')
ecplx('5.3518479027559984754E-1')

Voigt Profile distribution, pdf#

math53.voigt_profile_pdf(q, a, b)#

Returns \(\text{pdf}(x)\), the probability density function of a random variable \(X\), following a Voigt Profile distribution with shape \(a > 0\), scale \(b > 0\), and the support interval \((0, +\infty)\). See also Wikipedia [1221].

\[\text{pdf}(x) = V(x; \sigma, \gamma) = \frac{\Re[w(z)]}{\sigma \sqrt{2 \pi}}, \quad \text{where } z = \frac{x + i\gamma}{\sigma \sqrt{\pi}}.\]

An example in Python

>>> from xlcalcnet import ecplx
>>> ecplx.VoigtProfilePdf(0.5)
ecplx('5.2359877559829887307E-1')
>>> ecplx.VoigtProfilePdf('0.1')
ecplx('5.3518479027559984754E-1')

Contents

Error function, and related functions#

Exponentially scaled complementary error function, \(\mathrm{erfcx}(x)\)#

Difference of error functions, \(\mathrm{erfh}(x,h) = \mathrm{erf}(x+h)-\mathrm{erf}(x-h)\)#

Difference of error functions, \(\mathrm{erf2}(x_1,x_2) = \mathrm{erf}(x_2)-\mathrm{erf}(x_1)\)#

Probability function \(Q(x) = \Phi(-x)\)#

Inverse of the exponentially scaled complementary error function, \(\mathrm{erfcx}^{-1}(x)\)#

Inverse of the imaginary error function, \(\mathrm{erfi}^{-1}(x)\)#

Generalized Dawson integral, \(F(p, x)\)#

Generalized error function, \(\mathrm{erfg}(p, x)\)#

Expint3, \(\mathrm{erfg}(3, x)\)#

Scaled repeated integrals of the complementary error function, \(i^n \mathrm{erfc}(x)\)#

Fresnel auxiliary function \(\mathrm{f}(x)\)#

Fresnel auxiliary function \(\mathrm{g}(x)\)#

Goodwin-Staton integral, \(G(x)\)#

Voigt function U#

Voigt function V#

Voigt function H#

Voigt Profile distribution, pdf#