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regrouped function list accoring to DLMF chapters #176

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regrouped function list accoring to DLMF chapters #176

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75 changes: 41 additions & 34 deletions docs/src/functions_overview.md
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# Functions
Here the *Special Functions* are listed according to the structure of [NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/).

## Error Functions, Dawson’s and Fresnel Integrals

## [Gamma Function](https://dlmf.nist.gov/5)
| Function | Description |
|:-------- |:----------- |
| [`gamma(z)`](@ref SpecialFunctions.gamma) | [gamma function](https://en.wikipedia.org/wiki/Gamma_function) ``\Gamma(z)`` |
| [`digamma(x)`](@ref SpecialFunctions.digamma) | [digamma function](https://en.wikipedia.org/wiki/Digamma_function) (i.e. the derivative of `lgamma` at `x`) |
| [`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) |
| [`trigamma(x)`](@ref SpecialFunctions.trigamma) | [trigamma function](https://en.wikipedia.org/wiki/Trigamma_function) (i.e the logarithmic second derivative of `gamma` at `x`) |
| [`polygamma(m,x)`](@ref SpecialFunctions.polygamma) | [polygamma function](https://en.wikipedia.org/wiki/Polygamma_function) (i.e the (m+1)-th derivative of the `lgamma` function at `x`) |
| [`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)) |
| [`beta_inc(a,b,x,y)`](@ref SpecialFunctions.beta_inc) | [incomplete beta function ratio Ix(a,b) and Iy(a,b)](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) (i.e evaluates Ix(a,b) and Iy(a,b) and returns tuple (p,q)) |
| [`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 |
| [`loggamma(x)`](@ref SpecialFunctions.loggamma) | accurate `log(gamma(x))` for large `x` |
| [`logabsgamma(x)`](@ref SpecialFunctions.logabsgamma) | accurate `log(abs(gamma(x)))` for large `x` |
| [`lgamma(x)`](@ref SpecialFunctions.lgamma) | accurate `log(gamma(x))` for large `x` |
| [`logfactorial(x)`](@ref SpecialFunctions.logfactorial) | accurate `log(factorial(x))` for large `x`; same as `lgamma(x+1)` for `x > 1`, zero otherwise |
| [`beta(x,y)`](@ref SpecialFunctions.beta) | [beta function](https://en.wikipedia.org/wiki/Beta_function) at `x,y` |
| [`logbeta(x,y)`](@ref SpecialFunctions.logbeta) | accurate `log(beta(x,y))` for large `x` or `y` |
| [`logabsbeta(x,y)`](@ref SpecialFunctions.logabsbeta) | accurate `log(abs(beta(x,y)))` for large `x` or `y` |
| [`logabsbinomial(x,y)`](@ref SpecialFunctions.logabsbinomial) | accurate `log(abs(beta(x,y)))` for large `x` or `y` |


## [Trigonometric Integrals](https://dlmf.nist.gov/6)
| Function | Description |
|:-------- |:----------- |
| [`sinint(x)`](@ref SpecialFunctions.sinint) | [sine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral) ``Si(x)`` |
| [`cosint(x)`](@ref SpecialFunctions.cosint) | [cosine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral) ``Ci(x)`` |


## [Error Functions, Dawson’s and Fresnel Integrals](https://dlmf.nist.gov/7)
| Function | Description |
|:-------- |:----------- |
| [`erf(x)`](@ref SpecialFunctions.erf) | [error function](https://en.wikipedia.org/wiki/Error_function) at ``x`` |
Expand All @@ -14,27 +42,18 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
| [`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`` |


## Elliptic Integrals

| Function | Description |
|:-------- |:----------- |
| [`ellipk(m)`](@ref SpecialFunctions.ellipk) | [complete elliptic integral of 1st kind](https://en.wikipedia.org/wiki/Elliptic_integral#Notational_variants) ``K(m)`` |
| [`ellipe(m)`](@ref SpecialFunctions.ellipe) | [complete elliptic integral of 2nd kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind) ``E(m)`` |

## Trigonometric Integrals
| Function | Description |
|:-------- |:----------- |
| [`sinint(x)`](@ref SpecialFunctions.sinint) | [sine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral) ``Si(x)`` |
| [`cosint(x)`](@ref SpecialFunctions.cosint) | [cosine integral](https://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral) ``Ci(x)`` |

## Airy, Bessel and Hankel Functions
## [Airy and Related Functions](https://dlmf.nist.gov/9)
| Function | Description |
|:-------- |:----------- |
| [`airyai(z)`](@ref SpecialFunctions.airyai) | [Airy Ai function](https://en.wikipedia.org/wiki/Airy_function) at `z` |
| [`airyaiprime(z)`](@ref SpecialFunctions.airyaiprime) | derivative of the Airy Ai function at `z` |
| [`airybi(z)`](@ref SpecialFunctions.airybi) | [Airy Bi function](https://en.wikipedia.org/wiki/Airy_function) at `z` |
| [`airybiprime(z)`](@ref SpecialFunctions.airybiprime) | derivative of the Airy Bi function at `z` |
| [`airyaix(z)`](@ref SpecialFunctions.airyaix), [`airyaiprimex(z)`](@ref SpecialFunctions.airyaiprimex), [`airybix(z)`](@ref SpecialFunctions.airybix), [`airybiprimex(z)`](@ref SpecialFunctions.airybiprimex) | scaled Airy Ai function and `k`th derivatives at `z` |
| [`airyaix(z)`](@ref SpecialFunctions.airyaix), [`airyaiprimex(z)`](@ref SpecialFunctions.airyaiprimex), [`airybix(z)`](@ref SpecialFunctions.airybix), [`airybiprimex(z)`](@ref SpecialFunctions.airybiprimex) | scaled Airy Ai function and `k`th derivatives at `z` |

## [Bessel Functions](https://dlmf.nist.gov/10)
| Function | Description |
|:-------- |:----------- |
| [`besselj(nu,z)`](@ref SpecialFunctions.besselj) | [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind of order `nu` at `z` |
| [`besselj0(z)`](@ref SpecialFunctions.besselj0) | `besselj(0,z)` |
| [`besselj1(z)`](@ref SpecialFunctions.besselj1) | `besselj(1,z)` |
Expand All @@ -53,27 +72,15 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
| [`besselk(nu,z)`](@ref SpecialFunctions.besselk) | modified [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind of order `nu` at `z` |
| [`besselkx(nu,z)`](@ref SpecialFunctions.besselkx) | scaled modified Bessel function of the second kind of order `nu` at `z` |

## Gamma and related functions

## [Elliptic Integrals](https://dlmf.nist.gov/19)
| Function | Description |
|:-------- |:----------- |
| [`gamma(z)`](@ref SpecialFunctions.gamma) | [gamma function](https://en.wikipedia.org/wiki/Gamma_function) ``\Gamma(z)`` |
| [`digamma(x)`](@ref SpecialFunctions.digamma) | [digamma function](https://en.wikipedia.org/wiki/Digamma_function) (i.e. the derivative of `lgamma` at `x`) |
| [`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) |
| [`trigamma(x)`](@ref SpecialFunctions.trigamma) | [trigamma function](https://en.wikipedia.org/wiki/Trigamma_function) (i.e the logarithmic second derivative of `gamma` at `x`) |
| [`polygamma(m,x)`](@ref SpecialFunctions.polygamma) | [polygamma function](https://en.wikipedia.org/wiki/Polygamma_function) (i.e the (m+1)-th derivative of the `lgamma` function at `x`) |
| [`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)) |
| [`beta_inc(a,b,x,y)`](@ref SpecialFunctions.beta_inc) | [incomplete beta function ratio Ix(a,b) and Iy(a,b)](https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function) (i.e evaluates Ix(a,b) and Iy(a,b) and returns tuple (p,q)) |
| [`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 |
| [`loggamma(x)`](@ref SpecialFunctions.loggamma) | accurate `log(gamma(x))` for large `x` |
| [`logabsgamma(x)`](@ref SpecialFunctions.logabsgamma) | accurate `log(abs(gamma(x)))` for large `x` |
| [`lgamma(x)`](@ref SpecialFunctions.lgamma) | accurate `log(gamma(x))` for large `x` |
| [`logfactorial(x)`](@ref SpecialFunctions.logfactorial) | accurate `log(factorial(x))` for large `x`; same as `lgamma(x+1)` for `x > 1`, zero otherwise |
| [`beta(x,y)`](@ref SpecialFunctions.beta) | [beta function](https://en.wikipedia.org/wiki/Beta_function) at `x,y` |
| [`logbeta(x,y)`](@ref SpecialFunctions.logbeta) | accurate `log(beta(x,y))` for large `x` or `y` |
| [`logabsbeta(x,y)`](@ref SpecialFunctions.logabsbeta) | accurate `log(abs(beta(x,y)))` for large `x` or `y` |
| [`logabsbinomial(x,y)`](@ref SpecialFunctions.logabsbinomial) | accurate `log(abs(beta(x,y)))` for large `x` or `y` |
| [`ellipk(m)`](@ref SpecialFunctions.ellipk) | [complete elliptic integral of 1st kind](https://en.wikipedia.org/wiki/Elliptic_integral#Notational_variants) ``K(m)`` |
| [`ellipe(m)`](@ref SpecialFunctions.ellipe) | [complete elliptic integral of 2nd kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind) ``E(m)`` |


## Riemann Zeta and related functions
## [Zeta and Related Functions](https://dlmf.nist.gov/25)
| Function | Description |
|:-------- |:----------- |
| [`eta(x)`](@ref SpecialFunctions.eta) | [Dirichlet eta function](https://en.wikipedia.org/wiki/Dirichlet_eta_function) at `x` |
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