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Replace TeX escape sequences with unicode counterparts
The default KaTeX renderer supports it, and it also clears up the docstrings when viewing in the REPL
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docs/src/functions_overview.md

Lines changed: 8 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -8,15 +8,15 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
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99
| Function | Description |
1010
|:-------- |:----------- |
11-
| [`gamma(z)`](@ref SpecialFunctions.gamma(::Number)) | [gamma function](https://en.wikipedia.org/wiki/Gamma_function) ``\Gamma(z)`` |
11+
| [`gamma(z)`](@ref SpecialFunctions.gamma(::Number)) | [gamma function](https://en.wikipedia.org/wiki/Gamma_function) ``Γ(z)`` |
1212
| [`loggamma(x)`](@ref SpecialFunctions.loggamma(::Number)) | accurate `log(gamma(x))` for large `x` |
1313
| [`logabsgamma(x)`](@ref SpecialFunctions.logabsgamma) | accurate `log(abs(gamma(x)))` for large `x` |
1414
| [`logfactorial(x)`](@ref SpecialFunctions.logfactorial) | accurate `log(factorial(x))` for large `x`; same as `loggamma(x+1)` for `x > 1`, zero otherwise |
1515
| [`digamma(x)`](@ref SpecialFunctions.digamma) | [digamma function](https://en.wikipedia.org/wiki/Digamma_function) (i.e. the derivative of `loggamma` at `x`) |
1616
| [`invdigamma(x)`](@ref SpecialFunctions.invdigamma) | [invdigamma function](http://bariskurt.com/calculating-the-inverse-of-digamma-function/) (i.e. inverse of `digamma` function at `x` using fixed-point iteration algorithm) |
1717
| [`trigamma(x)`](@ref SpecialFunctions.trigamma) | [trigamma function](https://en.wikipedia.org/wiki/Trigamma_function) (i.e the logarithmic second derivative of `gamma` at `x`) |
1818
| [`polygamma(m,x)`](@ref SpecialFunctions.polygamma) | [polygamma function](https://en.wikipedia.org/wiki/Polygamma_function) (i.e the (m+1)-th derivative of the `loggamma` function at `x`) |
19-
| [`gamma(a,z)`](@ref SpecialFunctions.gamma(::Number,::Number)) | [upper incomplete gamma function ``\Gamma(a,z)``](https://en.wikipedia.org/wiki/Incomplete_gamma_function) |
19+
| [`gamma(a,z)`](@ref SpecialFunctions.gamma(::Number,::Number)) | [upper incomplete gamma function ``Γ(a,z)``](https://en.wikipedia.org/wiki/Incomplete_gamma_function) |
2020
| [`loggamma(a,z)`](@ref SpecialFunctions.loggamma(::Number,::Number)) | accurate `log(gamma(a,x))` for large arguments |
2121
| [`gamma_inc(a,x,IND)`](@ref SpecialFunctions.gamma_inc) | [incomplete gamma function ratio P(a,x) and Q(a,x)](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates P(a,x) and Q(a,x) for accuracy specified by IND and returns tuple (p,q)) |
2222
| [`gamma_inc_inv(a,p,q)`](@ref SpecialFunctions.gamma_inc_inv) | [inverse of incomplete gamma function ratio P(a,x) and Q(a,x)](https://en.wikipedia.org/wiki/Incomplete_gamma_function) (i.e evaluates x given P(a,x)=p and Q(a,x)=q) |
@@ -34,11 +34,11 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
3434

3535
| Function | Description |
3636
|:-------- |:----------- |
37-
| [`expint(ν, z)`](@ref SpecialFunctions.expint) | [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral) ``\operatorname{E}_\nu(z)`` |
38-
| [`expinti(x)`](@ref SpecialFunctions.expinti) | [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral) ``\operatorname{Ei}(x)`` |
39-
| [`expintx(x)`](@ref SpecialFunctions.expintx) | scaled [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral) ``e^z \operatorname{E}_\nu(z)`` |
40-
| [`sinint(x)`](@ref SpecialFunctions.sinint) | [sine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral) ``\operatorname{Si}(x)`` |
41-
| [`cosint(x)`](@ref SpecialFunctions.cosint) | [cosine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral) ``\operatorname{Ci}(x)`` |
37+
| [`expint(ν, z)`](@ref SpecialFunctions.expint) | [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral) ``\operatorname{E}(z)`` |
38+
| [`expinti(x)`](@ref SpecialFunctions.expinti) | [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral) ``\operatorname{Ei}(x)`` |
39+
| [`expintx(x)`](@ref SpecialFunctions.expintx) | scaled [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral) ``e^z \operatorname{E}(z)`` |
40+
| [`sinint(x)`](@ref SpecialFunctions.sinint) | [sine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral) ``\operatorname{Si}(x)`` |
41+
| [`cosint(x)`](@ref SpecialFunctions.cosint) | [cosine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral) ``\operatorname{Ci}(x)`` |
4242

4343

4444
## Error Functions, Dawson’s and Fresnel Integrals
@@ -56,7 +56,7 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
5656
| [`logerfcx(x)`](@ref SpecialFunctions.logerfcx) | log of the scaled complementary error function, i.e. accurate ``\operatorname{ln}(\operatorname{erfcx}(x))`` for large negative ``x`` |
5757
| [`erfi(x)`](@ref SpecialFunctions.erfi) | imaginary error function defined as ``-i \operatorname{erf}(ix)`` |
5858
| [`erfinv(x)`](@ref SpecialFunctions.erfinv) | inverse function to [`erf()`](@ref SpecialFunctions.erf) |
59-
| [`dawson(x)`](@ref SpecialFunctions.dawson) | scaled imaginary error function, a.k.a. Dawson function, i.e. accurate ``\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)`` for large ``x`` |
59+
| [`dawson(x)`](@ref SpecialFunctions.dawson) | scaled imaginary error function, a.k.a. Dawson function, i.e. accurate ``\frac{\sqrt{π}}{2} e^{-x^2} \operatorname{erfi}(x)`` for large ``x`` |
6060
| [`faddeeva(x)`](@ref SpecialFunctions.faddeeva) | [Faddeeva function](https://en.wikipedia.org/wiki/Faddeeva_function), equivalent to ``\operatorname{erfcx}(-ix)`` |
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6262

src/bessel.jl

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Original file line numberDiff line numberDiff line change
@@ -465,7 +465,7 @@ end
465465
@doc raw"""
466466
besseli(nu, x)
467467
468-
Modified Bessel function of the first kind of order `nu`, ``I_\nu(x)``.
468+
Modified Bessel function of the first kind of order `nu`, ``I_ν(x)``.
469469
470470
External links:
471471
[DLMF 10.25.2](https://dlmf.nist.gov/10.25.2),
@@ -486,7 +486,7 @@ end
486486
@doc raw"""
487487
besselix(nu, x)
488488
489-
Scaled modified Bessel function of the first kind of order `nu`, ``I_\nu(x) e^{- | \operatorname{Re}(x) |}``.
489+
Scaled modified Bessel function of the first kind of order `nu`, ``I_ν(x) e^{- | \operatorname{Re}(x) |}``.
490490
491491
External links:
492492
[DLMF 10.25.2](https://dlmf.nist.gov/10.25.2),
@@ -507,7 +507,7 @@ end
507507
@doc raw"""
508508
besselj(nu, x)
509509
510-
Bessel function of the first kind of order `nu`, ``J_\nu(x)``.
510+
Bessel function of the first kind of order `nu`, ``J_ν(x)``.
511511
512512
External links:
513513
[DLMF 10.2.2](https://dlmf.nist.gov/10.2.2),
@@ -532,7 +532,7 @@ end
532532
@doc raw"""
533533
besseljx(nu, x)
534534
535-
Scaled Bessel function of the first kind of order `nu`, ``J_\nu(x) e^{- | \operatorname{Im}(x) |}``.
535+
Scaled Bessel function of the first kind of order `nu`, ``J_ν(x) e^{- | \operatorname{Im}(x) |}``.
536536
537537
External links:
538538
[DLMF 10.2.2](https://dlmf.nist.gov/10.2.2),
@@ -553,7 +553,7 @@ end
553553
@doc raw"""
554554
besselk(nu, x)
555555
556-
Modified Bessel function of the second kind of order `nu`, ``K_\nu(x)``.
556+
Modified Bessel function of the second kind of order `nu`, ``K_ν(x)``.
557557
558558
External links:
559559
[DLMF 10.25.3](https://dlmf.nist.gov/10.25.3),
@@ -576,7 +576,7 @@ end
576576
@doc raw"""
577577
besselkx(nu, x)
578578
579-
Scaled modified Bessel function of the second kind of order `nu`, ``K_\nu(x) e^x``.
579+
Scaled modified Bessel function of the second kind of order `nu`, ``K_ν(x) e^x``.
580580
581581
External links:
582582
[DLMF 10.25.3](https://dlmf.nist.gov/10.25.3),
@@ -599,7 +599,7 @@ end
599599
"""
600600
bessely(nu, x)
601601
602-
Bessel function of the second kind of order `nu`, ``Y_\\nu(x)``.
602+
Bessel function of the second kind of order `nu`, ``Y_ν(x)``.
603603
604604
External links:
605605
[DLMF 10.2.3](https://dlmf.nist.gov/10.2.3),
@@ -620,7 +620,7 @@ end
620620
besselyx(nu, x)
621621
622622
Scaled Bessel function of the second kind of order `nu`,
623-
``Y_\\nu(x) e^{- | \\operatorname{Im}(x) |}``.
623+
``Y_ν(x) e^{- | \\operatorname{Im}(x) |}``.
624624
625625
External links:
626626
[DLMF 10.2.3](https://dlmf.nist.gov/10.2.3),
@@ -784,7 +784,7 @@ sphericalbessely(nu, x::T) where {T} = √((float(T))(π)/2x) * bessely(nu + one
784784
"""
785785
hankelh1(nu, x)
786786
787-
Bessel function of the third kind of order `nu`, ``H^{(1)}_\\nu(x)``.
787+
Bessel function of the third kind of order `nu`, ``H^{(1)}(x)``.
788788
789789
External links:
790790
[DLMF 10.2.5](https://dlmf.nist.gov/10.2.5),
@@ -797,7 +797,7 @@ hankelh1(nu, z) = besselh(nu, 1, z)
797797
"""
798798
hankelh2(nu, x)
799799
800-
Bessel function of the third kind of order `nu`, ``H^{(2)}_\\nu(x)``.
800+
Bessel function of the third kind of order `nu`, ``H^{(2)}(x)``.
801801
802802
External links:
803803
[DLMF 10.2.6](https://dlmf.nist.gov/10.2.6),
@@ -810,7 +810,7 @@ hankelh2(nu, z) = besselh(nu, 2, z)
810810
"""
811811
hankelh1x(nu, x)
812812
813-
Scaled Bessel function of the third kind of order `nu`, ``H^{(1)}_\\nu(x) e^{-x i}``.
813+
Scaled Bessel function of the third kind of order `nu`, ``H^{(1)}(x) e^{-x i}``.
814814
815815
External links:
816816
[DLMF 10.2.5](https://dlmf.nist.gov/10.2.5),
@@ -823,7 +823,7 @@ hankelh1x(nu, z) = besselhx(nu, 1, z)
823823
@doc raw"""
824824
hankelh2x(nu, x)
825825
826-
Scaled Bessel function of the third kind of order `nu`, ``H^{(2)}_\nu(x) e^{x i}``.
826+
Scaled Bessel function of the third kind of order `nu`, ``H^{(2)}(x) e^{x i}``.
827827
828828
External links:
829829
[DLMF 10.2.6](https://dlmf.nist.gov/10.2.6),
@@ -839,7 +839,7 @@ hankelh2x(nu, z) = besselhx(nu, 2, z)
839839
Bessel function of the first kind divided by `x`.
840840
Following convention:
841841
```math
842-
\operatorname{jinc}{x} = \frac{2 J_1({\pi x})}{\pi x}.
842+
\operatorname{jinc}{x} = \frac{2 J_1({π x})}{π x}.
843843
```
844844
Sometimes known as sombrero or besinc function.
845845

src/beta_inc.jl

Lines changed: 8 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -7,7 +7,7 @@ const exparg_p = log(prevfloat(floatmax(Float64)))
77
@doc raw"""
88
loggammadiv(a,b)
99
10-
Computes ``\log(\Gamma(b)/\Gamma(a+b))`` when `b >= 8`
10+
Computes ``\log(Γ(b)/Γ(a+b))`` when `b >= 8`
1111
"""
1212
loggammadiv(a::Number, b::Number) = _loggammadiv(promote(float(a), float(b))...)
1313

@@ -76,7 +76,7 @@ end
7676
@doc raw"""
7777
esum(mu,x)
7878
79-
Compute ``e^{\mu+x}``
79+
Compute ``e^{μ+x}``
8080
"""
8181
function esum(mu::Float64, x::Float64)
8282
if x > 0.0
@@ -95,7 +95,7 @@ end
9595
@doc raw"""
9696
beta_integrand(a, b, x, y, mu=0.0)
9797
98-
Compute ``e^{\mu} x^a y^b / B(a,b)``
98+
Compute ``e^{μ} x^a y^b / B(a,b)``
9999
"""
100100
function beta_integrand(a::Float64, b::Float64, x::Float64, y::Float64, mu::Float64=0.0)
101101
a0, b0 = minmax(a,b)
@@ -181,7 +181,7 @@ end
181181
beta_inc_cont_fraction(a,b,x,y,lambda,epps)
182182
183183
Compute ``I_{x}(a,b)`` using continued fraction expansion when `a, b > 1`.
184-
It is assumed that ``\lambda = (a+b)*y - b``
184+
It is assumed that ``λ = (a+b)*y - b``
185185
186186
External links:
187187
[DLMF 8.17.22](https://dlmf.nist.gov/8.17.22),
@@ -251,7 +251,7 @@ end
251251
beta_inc_asymptotic_symmetric(a,b,lambda,epps)
252252
253253
Compute ``I_{x}(a,b)`` using asymptotic expansion for `a, b >= 15`.
254-
It is assumed that ``\lambda = (a+b)*y - b``.
254+
It is assumed that ``λ = (a+b)*y - b``.
255255
256256
External links:
257257
[DLMF 8.17.22](https://dlmf.nist.gov/8.17.22),
@@ -525,7 +525,7 @@ end
525525
526526
Computes ``I_x(a,b)`` using power series:
527527
```math
528-
I_{x}(a,b) = G(a,b) x^{a}/a \left[1 + a \sum_{j=1}^{\infty} ((1-b)(2-b)\dots(j-b)/j!(a+j)) x^{j}\right]
528+
I_{x}(a,b) = G(a,b) x^{a}/a \left[1 + a \sum_{j=1}^ ((1-b)(2-b)\dots(j-b)/j!(a+j)) x^{j}\right]
529529
```
530530
External links:
531531
[DLMF 8.17.22](https://dlmf.nist.gov/8.17.22),
@@ -733,9 +733,9 @@ end
733733
Return a tuple ``(I_{x}(a,b), 1-I_{x}(a,b))`` where ``I_{x}(a,b)`` is the regularized
734734
incomplete beta function given by
735735
```math
736-
I_{x}(a,b) = \frac{1}{B(a,b)} \int_{0}^{x} t^{a-1}(1-t)^{b-1} \mathrm{d}t,
736+
I_{x}(a,b) = \frac{1}{B(a,b)} \int_0^x t^{a-1}(1-t)^{b-1} \mathrm{d}t,
737737
```
738-
where ``B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)``.
738+
where ``B(a,b) = Γ(a)Γ(b)/Γ(a+b)``.
739739
740740
External links:
741741
[DLMF 8.17.1](https://dlmf.nist.gov/8.17.1),

src/betanc.jl

Lines changed: 11 additions & 11 deletions
Original file line numberDiff line numberDiff line change
@@ -10,9 +10,9 @@ const errmax = 1e-15
1010
Compute tail of the noncentral beta distribution.
1111
Uses the recursive relation
1212
```math
13-
I_{x}(a,b+1;0) = I_{x}(a,b;0) - \Gamma(a+b)/\Gamma(a+1)\Gamma(b) x^a (1-x)^b
13+
I_{x}(a,b+1;0) = I_{x}(a,b;0) - Γ(a+b)/Γ(a+1)Γ(b) x^a (1-x)^b
1414
```
15-
and ``\Gamma(a+1) = a\Gamma(a)`` given in [DLMF 8.17.21](https://dlmf.nist.gov/8.17.21).
15+
and ``Γ(a+1) = (a)`` given in [DLMF 8.17.21](https://dlmf.nist.gov/8.17.21).
1616
"""
1717
function ncbeta_tail(a::Float64, b::Float64, lambda::Float64, x::Float64)
1818
if x <= 0.0
@@ -55,16 +55,16 @@ end
5555
ncbeta_poisson(a,b,lambda,x)
5656
5757
Compute CDF of noncentral beta if `lambda >= 54` using:
58-
First ``\lambda/2`` is calculated and the Poisson term is calculated using
59-
``P(j-1) = j/\lambda P(j)`` and ``P(j+1) = \lambda/(j+1) P(j)``.
58+
First ``λ/2`` is calculated and the Poisson term is calculated using
59+
``P(j-1) = j/λ P(j)`` and ``P(j+1) = λ/(j+1) P(j)``.
6060
Then backward recurrences are used until either the Poisson weights fall below
6161
`errmax` or `iterlo` is reached.
6262
```math
63-
I_{x}(a+j-1,b) = I_{x}(a+j,b) + \Gamma(a+b+j-1)/\Gamma(a+j)\Gamma(b)x^{a+j-1}(1-x)^{b}
63+
I_{x}(a+j-1,b) = I_{x}(a+j,b) + Γ(a+b+j-1)/Γ(a+j)Γ(b) x^{a+j-1} (1-x)^{b}
6464
```
6565
Then forward recurrences are used until error bound falls below `errmax`.
6666
```math
67-
I_{x}(a+j+1,b) = I_{x}(a+j,b) - \Gamma(a+b+j)/\Gamma(a+j)\Gamma(b)x^{a+j}(1-x)^{b}
67+
I_{x}(a+j+1,b) = I_{x}(a+j,b) - Γ(a+b+j)/Γ(a+j)Γ(b) x^{a+j} (1-x)^{b}
6868
```
6969
"""
7070
function ncbeta_poisson(a::Float64, b::Float64, lambda::Float64, x::Float64)
@@ -147,10 +147,10 @@ end
147147
148148
Compute the CDF of the noncentral beta distribution given by
149149
```math
150-
I_{x}(a,b; \lambda) = \sum_{j=0}^{\infty} q(\lambda/2,j) I_{x}(a+j,b;0)
150+
I_{x}(a,b; λ) = \sum_{j=0}^ q(λ/2,j) I_{x}(a+j,b;0)
151151
```
152-
For ``\lambda < 54`` : algorithm suggested by Lenth(1987) in `ncbeta_tail(a,b,lambda,x)`.
153-
Else for ``\lambda \geq 54``: modification in Chattamvelli(1997) in
152+
For ``λ < 54`` : algorithm suggested by Lenth(1987) in `ncbeta_tail(a,b,lambda,x)`.
153+
Else for ``λ ≥ 54``: modification in Chattamvelli(1997) in
154154
`ncbeta_poisson(a,b,lambda,x)` by using both forward and backward recurrences.
155155
"""
156156
function ncbeta(a::Float64, b::Float64, lambda::Float64, x::Float64)
@@ -173,9 +173,9 @@ end
173173
174174
Compute CDF of noncentral F distribution given by:
175175
```math
176-
F(x, v_1, v_2; \lambda) = I_{v_1 x/(v_1 x + v_2)}(v_1/2, v_2/2; \lambda)
176+
F(x, v_1, v_2; λ) = I_{v_1 x/(v_1 x + v_2)}(v_1/2, v_2/2; λ)
177177
```
178-
where ``I_{x}(a,b; \lambda)`` is the noncentral beta function computed above.
178+
where ``I_{x}(a,b; λ)`` is the noncentral beta function computed above.
179179
180180
External links:
181181
[Wikipedia](https://en.wikipedia.org/wiki/Noncentral_F-distribution)

src/ellip.jl

Lines changed: 10 additions & 10 deletions
Original file line numberDiff line numberDiff line change
@@ -11,8 +11,8 @@ Computes Complete Elliptic Integral of 1st kind ``K(m)`` for parameter ``m`` giv
1111
```math
1212
\operatorname{ellipk}(m)
1313
= K(m)
14-
= \int_0^{ \frac{\pi}{2} } \frac{1}{\sqrt{1 - m \sin^2 \theta}} \, \mathrm{d}\theta
15-
\quad \text{for} \quad m \in \left( -\infty, 1 \right] \, .
14+
= \int_0^{ \frac{π}{2} } \frac{1}{\sqrt{1 - m \sin^2 θ}} \, \mathrm{d}θ
15+
\quad \text{for} \quad m \in \left( -, 1 \right] \, .
1616
```
1717
1818
External links:
@@ -22,9 +22,9 @@ External links:
2222
See also: [`ellipe(m)`](@ref SpecialFunctions.ellipe).
2323
2424
# Arguments
25-
- `m`: parameter ``m``, restricted to the domain ``(-\infty,1]``, is related to
25+
- `m`: parameter ``m``, restricted to the domain ``(-,1]``, is related to
2626
the elliptic modulus ``k`` by ``k^2=m`` and to the modular angle
27-
``\alpha`` by ``k = \sin \alpha``.
27+
``α`` by ``k = \sin α``.
2828
2929
# Implementation
3030
Using piecewise approximation polynomial as given in
@@ -40,7 +40,7 @@ For ``m<0``, followed by
4040
> Journal of Computational and Applied Mathematics. 282.
4141
> DOI 10.13140/2.1.1946.6245.,
4242
> <https://www.researchgate.net/publication/267330394>
43-
As suggested in this paper, the domain is restricted to ``(-\infty,1]``.
43+
As suggested in this paper, the domain is restricted to ``(-,1]``.
4444
"""
4545
ellipk(m::Real) = _ellipk(float(m))
4646

@@ -191,8 +191,8 @@ Computes Complete Elliptic Integral of 2nd kind ``E(m)`` for parameter ``m`` giv
191191
```math
192192
\operatorname{ellipe}(m)
193193
= E(m)
194-
= \int_0^{ \frac{\pi}{2} } \sqrt{1 - m \sin^2 \theta} \, \mathrm{d}\theta
195-
\quad \text{for} \quad m \in \left( -\infty, 1 \right] .
194+
= \int_0^{ \frac{π}{2} } \sqrt{1 - m \sin^2 θ} \, \mathrm{d}θ
195+
\quad \text{for} \quad m \in \left( -, 1 \right] .
196196
```
197197
198198
External links:
@@ -202,9 +202,9 @@ External links:
202202
See also: [`ellipk(m)`](@ref SpecialFunctions.ellipk).
203203
204204
# Arguments
205-
- `m`: parameter ``m``, restricted to the domain ``(-\infty,1]``, is related to
205+
- `m`: parameter ``m``, restricted to the domain ``(-,1]``, is related to
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the elliptic modulus ``k`` by ``k^2=m`` and to the modular angle
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``\alpha`` by ``k=\sin \alpha``.
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``α`` by ``k = \sin α``.
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# Implementation
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Using piecewise approximation polynomial as given in
@@ -220,7 +220,7 @@ For ``m<0``, followed by
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> Journal of Computational and Applied Mathematics. 282.
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> DOI 10.13140/2.1.1946.6245.,
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> <https://www.researchgate.net/publication/267330394>
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As suggested in this paper, the domain is restricted to ``(-\infty,1]``.
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As suggested in this paper, the domain is restricted to ``(-,1]``.
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"""
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ellipe(m::Real) = _ellipe(float(m))
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src/erf.jl

Lines changed: 4 additions & 4 deletions
Original file line numberDiff line numberDiff line change
@@ -82,7 +82,7 @@ end
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Compute the error function of ``x``, defined by
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```math
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\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-t^2) \; \mathrm{d}t
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\operatorname{erf}(x) = \frac{2}{\sqrt{π}} \int_0^x \exp(-t^2) \; \mathrm{d}t
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\quad \text{for} \quad x \in \mathbb{C} \, .
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```
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@@ -126,7 +126,7 @@ Compute the complementary error function of ``x``, defined by
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```math
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\operatorname{erfc}(x)
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= 1 - \operatorname{erf}(x)
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= \frac{2}{\sqrt{\pi}} \int_x^\infty \exp(-t^2) \; \mathrm{d}t
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= \frac{2}{\sqrt{π}} \int_x^ \exp(-t^2) \; \mathrm{d}t
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\quad \text{for} \quad x \in \mathbb{C} \, .
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```
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@@ -200,11 +200,11 @@ Compute the Dawson function (scaled imaginary error function) of ``x``, defined
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```math
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\operatorname{dawson}(x)
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= \frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)
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= \frac{\sqrt{π}}{2} e^{-x^2} \operatorname{erfi}(x)
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\quad \text{for} \quad x \in \mathbb{C} \, .
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```
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This is the accurate version of ``\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)``
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This is the accurate version of ``\frac{\sqrt{π}}{2} e^{-x^2} \operatorname{erfi}(x)``
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for large ``x``.
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210210
External links: [DLMF 7.2.5](https://dlmf.nist.gov/7.2.5),

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