update docstrings (#201)

* update docs for beta_inc

* tidy up links for gamma_inc, beta_inc

* links for bessel functions. small format for sinint, cosint

* cross and external links for bessel, hankel, airy

* small tidy ups in refs

* \operatornane is bugging Travis?

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* Update src/bessel.jl

Co-Authored-By: Simon Byrne <[email protected]>

* add couple of equation links to DLMF

* order of see also and links; ellipk/e clarifications

* order of links/see also

* range -> domain

* DLMF links now in X.Y.Z form

Co-authored-by: Simon Byrne <[email protected]>

Author: AshtonSBradley

src/bessel.jl

@@ -66,9 +66,9 @@ end
 
 Airy function of the first kind ``\\operatorname{Ai}(x)``.
 
-See also: [`airyaix`](@ref), [`airyaiprime`](@ref), [`airybi`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airyaix`](@ref), [`airyaiprime`](@ref), [`airybi`](@ref)
 """
 function airyai end
 airyai(z::Complex{Float64}) = _airy(z, Int32(0), Int32(1))
@@ -78,9 +78,9 @@ airyai(z::Complex{Float64}) = _airy(z, Int32(0), Int32(1))
 
 Derivative of the Airy function of the first kind ``\\operatorname{Ai}'(x)``.
 
-See also: [`airyaiprimex`](@ref), [`airyai`](@ref), [`airybi`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airyaiprimex`](@ref), [`airyai`](@ref), [`airybi`](@ref)
 """
 function airyaiprime end
 airyaiprime(z::Complex{Float64}) =  _airy(z, Int32(1), Int32(1))
@@ -90,9 +90,9 @@ airyaiprime(z::Complex{Float64}) =  _airy(z, Int32(1), Int32(1))
 
 Airy function of the second kind ``\\operatorname{Bi}(x)``.
 
-See also: [`airybix`](@ref), [`airybiprime`](@ref),  [`airyai`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airybix`](@ref), [`airybiprime`](@ref),  [`airyai`](@ref)
 """
 function airybi end
 airybi(z::Complex{Float64}) = _biry(z, Int32(0), Int32(1))
@@ -102,9 +102,9 @@ airybi(z::Complex{Float64}) = _biry(z, Int32(0), Int32(1))
 
 Derivative of the Airy function of the second kind ``\\operatorname{Bi}'(x)``.
 
-See also: [`airybiprimex`](@ref), [`airybi`](@ref), [`airyai`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airybiprimex`](@ref), [`airybi`](@ref), [`airyai`](@ref)
 """
 function airybiprime end
 airybiprime(z::Complex{Float64}) = _biry(z, Int32(1), Int32(1))
@@ -115,9 +115,9 @@ airybiprime(z::Complex{Float64}) = _biry(z, Int32(1), Int32(1))
 Scaled Airy function of the first kind ``\\operatorname{Ai}(x) e^{\\frac{2}{3} x
 \\sqrt{x}}``.  Throws `DomainError` for negative `Real` arguments.
 
-See also: [`airyai`](@ref), [`airyaiprime`](@ref), [`airybi`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airyai`](@ref), [`airyaiprime`](@ref), [`airybi`](@ref)
 """
 function airyaix end
 airyaix(z::Complex{Float64}) = _airy(z, Int32(0), Int32(2))
@@ -128,9 +128,9 @@ airyaix(z::Complex{Float64}) = _airy(z, Int32(0), Int32(2))
 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.
 
-See also: [`airyaiprime`](@ref), [`airyai`](@ref), [`airybi`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airyaiprime`](@ref), [`airyai`](@ref), [`airybi`](@ref)
 """
 function airyaiprimex end
 airyaiprimex(z::Complex{Float64}) =  _airy(z, Int32(1), Int32(2))
@@ -140,9 +140,9 @@ airyaiprimex(z::Complex{Float64}) =  _airy(z, Int32(1), Int32(2))
 
 Scaled Airy function of the second kind ``\\operatorname{Bi}(x) e^{- \\left| \\operatorname{Re} \\left( \\frac{2}{3} x \\sqrt{x} \\right) \\right|}``.
 
-See also: [`airybi`](@ref), [`airybiprime`](@ref), [`airyai`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airybi`](@ref), [`airybiprime`](@ref), [`airyai`](@ref)
 """
 function airybix end
 airybix(z::Complex{Float64}) = _biry(z, Int32(0), Int32(2))
@@ -152,9 +152,9 @@ airybix(z::Complex{Float64}) = _biry(z, Int32(0), Int32(2))
 
 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|}``.
 
-See also: [`airybiprime`](@ref), [`airybi`](@ref), [`airyai`](@ref)
-
 External links: [DLMF](https://dlmf.nist.gov/9.2), [Wikipedia](https://en.wikipedia.org/wiki/Airy_function)
+
+See also: [`airybiprime`](@ref), [`airybi`](@ref), [`airyai`](@ref)
 """
 function airybiprimex end
 airybiprimex(z::Complex{Float64}) = _biry(z, Int32(1), Int32(2))

src/beta_inc.jl

@@ -175,7 +175,7 @@ end
 Compute ``I_{x}(a,b)`` using continued fraction expansion when `a,b > 1`.
 It is assumed that ``\\lambda = (a+b)*y - b``
 
-External links: [DLMF](https://dlmf.nist.gov/8.17#E22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc`](@ref)
 
@@ -241,7 +241,7 @@ end
 Compute ``I_{x}(a,b)`` using asymptotic expansion for `a,b >= 15`.
 It is assumed that ``\\lambda = (a+b)*y - b``.
 
-External links: [DLMF](https://dlmf.nist.gov/8.17#E22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc`](@ref)
 
@@ -340,7 +340,7 @@ end
 Evaluation of ``I_{x}(a,b)`` when `b < min(epps,epps*a)` and `x <= 0.5` using asymptotic expansion.
 It is assumed `a >= 15` and `b <= 1`, and epps is tolerance used.
 
-External links: [DLMF](https://dlmf.nist.gov/8.17#E22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc`](@ref)
 """
@@ -413,7 +413,7 @@ end
 
 Variant of `BPSER(A,B,X,EPS)`.
 
-External links: [DLMF](https://dlmf.nist.gov/8.17#E22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc`](@ref)
 
@@ -455,7 +455,7 @@ end
 
 Another variant of `BPSER(A,B,X,EPS)`.
 
-External links: [DLMF](https://dlmf.nist.gov/8.17#E22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc`](@ref)
 
@@ -495,7 +495,7 @@ Computes ``I_x(a,b)`` using power series :
 ```math
 I_{x}(a,b) = G(a,b)x^{a}/a (1 + a\\sum_{j=1}^{\\infty}((1-b)(2-b)...(j-b)/j!(a+j)) x^{j})
 ```
-External links: [DLMF](https://dlmf.nist.gov/8.17#E22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.22), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc`](@ref)
 
@@ -592,9 +592,9 @@ end
     beta_inc_diff(a,b,x,y,n,epps)
 
 Compute ``I_{x}(a,b) - I_{x}(a+n,b)`` where `n` is positive integer and epps is tolerance.
-A generalised version of [DLMF](https://dlmf.nist.gov/8.17#E20).
+A generalised version of [DLMF](https://dlmf.nist.gov/8.17.20).
 
-External links: [DLMF](https://dlmf.nist.gov/8.17#E20), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.20), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc`](@ref)
 """
@@ -699,7 +699,7 @@ I_{x}(a,b) = \\frac{1}{B(a,b)} \\int_{0}^{x} t^{a-1}(1-t)^{b-1} dt,
 ```
 where ``B(a,b) = \\Gamma(a)\\Gamma(b)/\\Gamma(a+b)``.
 
-External links: [DLMF](https://dlmf.nist.gov/8.17#E1), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
+External links: [DLMF](https://dlmf.nist.gov/8.17.1), [Wikipedia](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function)
 
 See also: [`beta_inc_inv(a,b,p,q)`](@ref SpecialFunctions.beta_inc_inv)
 """

src/ellip.jl

@@ -15,14 +15,13 @@ Computes Complete Elliptic Integral of 1st kind ``K(m)`` for parameter ``m`` giv
 \quad \text{for} \quad m \in \left( -\infty, 1 \right] \, .
 ```
 
-External links: [DLMF](https://dlmf.nist.gov/19.2#E8),
-[Wikipedia](https://en.wikipedia.org/wiki/Elliptic_integral#Notational_variants).
+External links: [DLMF](https://dlmf.nist.gov/19.2.8), [Wikipedia](https://en.wikipedia.org/wiki/Elliptic_integral#Notational_variants).
 
 See also: [`ellipe(m)`](@ref SpecialFunctions.ellipe).
 
 # Arguments
-- `m`: parameter ``m`` is in relation to elliptic modulus ``k`` by ``k^2=m`` and modular
-    angle ``\alpha`` by ``k=\sin \alpha``
+- `m`: parameter ``m``, restricted to the domain ``(-\infty,1]``, is related to the elliptic modulus ``k`` by ``k^2=m`` and to the modular
+    angle ``\alpha`` by ``k=\sin \alpha``.
 
 # Implementation
 Using piecewise approximation polynomial as given in
@@ -38,7 +37,7 @@ For ``m<0``, followed by
 > 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]``.
+As suggested in this paper, the domain is restricted to ``(-\infty,1]``.
 """
 function ellipk(m::Float64)
     flag_is_m_neg = false
@@ -191,14 +190,13 @@ Computes Complete Elliptic Integral of 2nd kind ``E(m)`` for parameter ``m`` giv
 \quad \text{for} \quad m \in \left( -\infty, 1 \right] \, .
 ```
 
-External links: [DLMF](https://dlmf.nist.gov/19.2#E8),
-[Wikipedia](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind).
+External links: [DLMF](https://dlmf.nist.gov/19.2.8), [Wikipedia](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind).
 
 See also: [`ellipk(m)`](@ref SpecialFunctions.ellipk).
 
 # Arguments
-- `m`: parameter ``m`` is in relation to elliptic modulus ``k`` by ``k^2=m`` and modular
-    angle ``\alpha`` by ``k=\sin \alpha``
+- `m`: parameter ``m``, restricted to the domain ``(-\infty,1]``, is related to the elliptic modulus ``k`` by ``k^2=m`` and to the modular
+    angle ``\alpha`` by ``k=\sin \alpha``.
 
 # Implementation
 Using piecewise approximation polynomial as given in
@@ -214,7 +212,7 @@ For ``m<0``, followed by
 > 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]``.
+As suggested in this paper, the domain is restricted to ``(-\infty,1]``.
 """
 function ellipe(m::Float64)
     flag_is_m_neg = false

src/erf.jl

@@ -51,7 +51,7 @@ Compute the error function of ``x``, defined by
 \quad \text{for} \quad x \in \mathbb{C} \, .
 ```
 
-External links: [DLMF](https://dlmf.nist.gov/7.2.E1), [Wikipedia](https://en.wikipedia.org/wiki/Error_function).
+External links: [DLMF](https://dlmf.nist.gov/7.2.1), [Wikipedia](https://en.wikipedia.org/wiki/Error_function).
 
 See also: [`erfc(x)`](@ref SpecialFunctions.erfc), [`erfcx(x)`](@ref SpecialFunctions.erfcx),
 [`erfi(x)`](@ref SpecialFunctions.erfi), [`dawson(x)`](@ref SpecialFunctions.dawson),
@@ -78,7 +78,7 @@ Compute the complementary error function of ``x``, defined by
 
 This is the accurate version of `1-erf(x)` for large ``x``.
 
-External links: [DLMF](https://dlmf.nist.gov/7.2.E2),
+External links: [DLMF](https://dlmf.nist.gov/7.2.2),
 [Wikipedia](https://en.wikipedia.org/wiki/Error_function#Complementary_error_function).
 
 See also: [`erf(x)`](@ref SpecialFunctions.erf).
@@ -104,7 +104,7 @@ Compute the scaled complementary error function of ``x``, defined by
 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](https://dlmf.nist.gov/7.2.E3),
+External links: [DLMF](https://dlmf.nist.gov/7.2.3),
 [Wikipedia](https://en.wikipedia.org/wiki/Error_function#Complementary_error_function).
 
 See also: [`erfc(x)`](@ref SpecialFunctions.erfc).
@@ -153,7 +153,7 @@ Compute the Dawson function (scaled imaginary error function) of ``x``, defined
 This is the accurate version of ``\frac{\sqrt{\pi}}{2} e^{-x^2} \operatorname{erfi}(x)``
 for large ``x``.
 
-External links: [DLMF](https://dlmf.nist.gov/7.2.E5),
+External links: [DLMF](https://dlmf.nist.gov/7.2.5),
 [Wikipedia](https://en.wikipedia.org/wiki/Dawson_function).
 
 See also: [`erfi(x)`](@ref SpecialFunctions.erfi).

src/gamma.jl

@@ -603,7 +603,7 @@ Compute the gamma function for complex ``z``, defined by
 and by analytic continuation in the whole complex plane.
 
 External links:
-[DLMF](https://dlmf.nist.gov/5.2.E1),
+[DLMF](https://dlmf.nist.gov/5.2.1),
 [Wikipedia](https://en.wikipedia.org/wiki/Gamma_function).
 
 See also: [`loggamma(z)`](@ref SpecialFunctions.loggamma).

src/gamma_inc.jl

@@ -358,7 +358,7 @@ R(a,x) * \\frac{1}{1-\\frac{z}{a+1+\\frac{z}{a+2-\\frac{(a+1)z}{a+3+\\frac{2z}{a
 ```
 Used when `1 <= a <= BIG` and `x < x0`.
 
-External links: [DLMF](https://dlmf.nist.gov/8.9#E2)
+External links: [DLMF](https://dlmf.nist.gov/8.9.2)
 
 See also: [`gamma_inc(a,x,ind)`](@ref SpecialFunctions.gamma_inc)
 """
@@ -397,7 +397,7 @@ R(a,x)/a * (1 + \\sum_{n=1}^{\\infty} x^{n}/((a+1)(a+2)...(a+n)))
 ```
 Used when `1 <= a <= BIG` and `x <= max{a, ln 10}`.
 
-External links: [DLMF](https://dlmf.nist.gov/8.11#E2)
+External links: [DLMF](https://dlmf.nist.gov/8.11.2)
 
 See also: [`gamma_inc(a,x,ind)`](@ref SpecialFunctions.gamma_inc)
 """
@@ -447,7 +447,7 @@ R(a,x)/a * (1 + \\sum_{n=1}^{N-1}(a_{n}/x^{n} + \\Theta _{n}a_{n}/x^{n}))
 ```
 where `R(a,x) = rgammax(a,x)`. Used when `1 <= a <= BIG` and `x >= x0`.
 
-External links: [DLMF](https://dlmf.nist.gov/8.11#E2)
+External links: [DLMF](https://dlmf.nist.gov/8.11.2)
 
 See also: [`gamma_inc(a,x,ind)`](@ref SpecialFunctions.gamma_inc)
 """
@@ -547,7 +547,7 @@ T(a,\\lambda) = \\sum_{0}^{N} c_{k}(z)a^{-k}
 This is a higher accuracy approximation of Temme expansion, which deals with the region near `a ≈ x` with `a` large.
 Refer Appendix F in the paper for the extensive set of coefficients calculated using Brent's multiple precision arithmetic(set at 50 digits) in BRENT, R. P. A FORTRAN multiple-precision arithmetic package, ACM Trans. Math. Softw. 4(1978), 57-70 .
 
-External links: [DLMF](https://dlmf.nist.gov/8.12#E8)
+External links: [DLMF](https://dlmf.nist.gov/8.12.8)
 
 See also: [`gamma_inc(a,x,ind)`](@ref SpecialFunctions.gamma_inc)
 """
@@ -608,7 +608,7 @@ T(a,\\lambda) = \\sum_{0}^{N} c_{k}(z)a^{-k}
 ```
 This mainly solves the problem near the region when `a ≈ x` with a large, and is a lower accuracy version of the minimax approximation.
 
-External links: [DLMF](https://dlmf.nist.gov/8.12#E8)
+External links: [DLMF](https://dlmf.nist.gov/8.12.8)
 
 See also: [`gamma_inc(a,x,ind)`](@ref SpecialFunctions.gamma_inc)
 """
@@ -642,7 +642,7 @@ E(y) = 1/2 - (1 - y/3)*(\\sqrt(y/\\pi))
 ```
 Used instead of it's previous function when ``\\sigma <= e_{0}/\\sqrt{a}``.
 
-External links: [DLMF](https://dlmf.nist.gov/8.12#E8)
+External links: [DLMF](https://dlmf.nist.gov/8.12.8)
 """
 function gamma_inc_temme_1(a::Float64, x::Float64, z::Float64, ind::Integer)
     iop = ind + 1
@@ -832,7 +832,7 @@ Q(x,a)=\\frac{1}{\\Gamma (a)} \\int_{x}^{\\infty} e^{-t}t^{a-1} dt.
 
 `IND ∈ [0,1,2]` sets accuracy: `IND=0` means 14 significant digits accuracy, `IND=1` means 6 significant digit, and `IND=2` means only 3 digit accuracy.
 
-External links: [DLMF](https://dlmf.nist.gov/8.2#E4), [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function)
+External links: [DLMF](https://dlmf.nist.gov/8.2.4), [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function)
 
 See also [`gamma(z)`](@ref SpecialFunctions.gamma), [`gamma_inc_inv(a,p,q)`](@ref SpecialFunctions.gamma_inc_inv)
 """
@@ -954,7 +954,7 @@ gamma_inc(a::AbstractFloat,x::AbstractFloat,ind::Integer) = throw(MethodError(ga
 
 Inverts the `gamma_inc(a,x)` function, by computing `x` given `a`,`p`,`q` in ``P(a,x)=p`` and ``Q(a,x)=q``.
 
-External links: [DLMF](https://dlmf.nist.gov/8.2#E4), [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function)
+External links: [DLMF](https://dlmf.nist.gov/8.2.4), [Wikipedia](https://en.wikipedia.org/wiki/Incomplete_gamma_function)
 
 See also: [`gamma_inc(a,x,ind)`](@ref SpecialFunctions.gamma_inc).
 """

src/sincosint.jl

@@ -238,7 +238,7 @@ Compute the sine integral function of ``x``, defined by
 x \in \mathbb{R} \,.
 ```
 
-External links: [DLMF](https://dlmf.nist.gov/6.2.E9),
+External links: [DLMF](https://dlmf.nist.gov/6.2.9),
 [Wikipedia](https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral).
 
 See also: [`cosint(x)`](@ref SpecialFunctions.cosint).
@@ -270,7 +270,7 @@ x > 0 \,,
 ```
 where ``\gamma`` is the Euler-Mascheroni constant.
 
-External links: [DLMF](https://dlmf.nist.gov/6.2.E13),
+External links: [DLMF](https://dlmf.nist.gov/6.2.13),
 [Wikipedia](https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral).
 
 See also: [`sinint(x)`](@ref SpecialFunctions.sinint).