The cephes Reference Manual

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The cephes Reference Manual

This is the cephes Reference Manual, version 1.2, generated automatically by Declt version 4.0 beta 2 "William Riker" on Thu Sep 15 03:26:06 2022 GMT+0.

Table of Contents


1 Introduction

Cephes Mathematical Library

A common lisp CFFI wrapper for the SciPy version of Cephes special functions.

For many years Cephes was considered the gold standard in cross-platform mathematical libraries, superior to the libm distributions of the time (early 1990's). Since then C99 and later ANSI C specifications have closed the gap, but there are still certain special and statistical functions available in Cephes that are not in the standard C libraries.

The SciPy version differs from standard Cephes in that it has some additional functions, improvements in accuracy, and is better documented. Only double-float versions are provided.

Installation

The ASDF file will automatically build the shared library as part of the load operation. If you need to build on a system other than MS Windows or UNIX, you will need to modify the make file to account for the linker command to create a shared library.

As delivered, the Makefile is set-up to work on MS Windows or UNIX, and you can build it manually like so:

cd scipy-cephes && make

If you build this on another platform, please drop a note into a Cephes repository issue with the build instructions so we can update the Makefile and system definition.

The init.lisp file, where CFFI loads the library, should work out of the box if libmd on the path somewhere, regardless of platform. If in doubt, place a copy in the same directory as this README.

Documentation

If you know your way around special functions, the table below should suffice to get started. For a more complete description, see the doc strings in cephes.lisp. Finally, the C source code itself, referenced in cephes.lisp, is thorough and complete from a mathematical perspective.

You can also use the scipy.special online documentation.

The API

There is no overlap between the wrapped Cephes functions and the Common Lisp numerical tower. All functions are in the cephes package.

Exported functions are:

| function | description | |-------------------|-------------------------------------------------------------------------------| | airy | Airy function | | bdtr | Binomial distribution | | bdtrc | Complement of binomial distribution | | bdtri | Inverse binomial distribution | | besselpoly | Weighted integral of the Bessel function of the first kind | | beta | Beta function | | lbeta | Natural log of beta | | btdtr | incomplete beta integral | | cbrt | cube root | | chdtr | Chi-square distribution | | chdtrc | Complemented Chi-square distribution | | chdtri | Inverse of complemented Chi-square distribution | | dawsn | Dawson's Integral | | ellik | Incomplete elliptic integral of the first kind | | ellie | Incomplete elliptic integral of the second kind | | ellpk | Complete elliptic integral of the first kind | | ellpe | Complete elliptic integral of the second kind | | jacobian-elliptic | jacobian Elliptic Functions | | exp2 | Base 2 exponential function | | exp10 | Base 10 exponential function (Common antilogarithm) | | expn | Exponential integral | | fdtr | F distribution | | fdtrc | Complemented F distribution | | fdtri | Inverse of F distribution | | fresnl | Fresnel integral | | gamma | Gamma function | | log-gamma | Natural logarithm of Gamma function | | gdtr | Gamma distribution function | | gdtrc | Complemented Gamma distribution function | | gdtri | Inverse Gamma distribution function (?) - not documented in src | | hyp2f1 | Gauss hypergeometric function | | hyperg | Confluent hypergeometric function | | i0 | Modified Bessel function of order zero | | i0e | Modified Bessel function of order zero, exponentially scaled | | i1 | Modified Bessel function of order one | | i1e | Modified Bessel function of order one, exponentially scaled | | igam | Regularized lower incomplete gamma function | | igamc | Regularized upper incomplete gamma function | | igami | Inverse of the lower incomplete gamma function | | igamci | Inverse of the upper incomplete gamma function | | incbet | Incomplete beta integral | | incbi | Inverse of incomplete beta integral | | iv | Modified Bessel function of noninteger order | | j0 | Bessel function of order zero | | y0 | Bessel function of the second kind, order zero | | j1 | Bessel function of order one | | y1 | Bessel function of second kind of order one | | jv | Bessel function of noninteger order | | k0 | Modified Bessel function, third kind, order zero | | k0e | Modified Bessel function, third kind, order zero, exponentially scaled | | k1 | Modified Bessel function of the third kind of order one | | k1e | Modified Bessel function of the third kind of order one, exponentially scaled | | kn | Modified Bessel function, third kind, integer order | | nbdtr | Negative binomial distribution | | nbdtrc | Complemented negative binomial distribution | | nbdtri | Inverse complemented negative binomial distribution | | ndtr | Normal distribution function | | log-ndtr | Log of the normal distribution function | | erf | Error function | | erfc | Complementary error function | | erfinv | Inverse of the error function | | erfcinv | Inverse of the complementary error function | | ndtri | Inverse of Normal distribution function | | pdtr | Poisson distribution | | pdtrc | Complemented poisson distribution | | pdtri | Inverse Poisson distribution | | poch | Pochhammer symbol (a)_m = gamma(a + m) / gamma(a) | | psi | Psi (digamma) function | | rgamma | Reciprocal Gamma function | | shichi | Hyperbolic sine and cosine integrals | | sici | Sine and cosine integrals | | sindg | Circular sine of angle in degrees | | cosdg | Circular cosine of angle in degrees | | sinpi | Compute sin(pi * x) | | cospi | Compute cos(pi * x) | | spence | Dilogarithm | | stdtr | Student's t distribution | | stdtri | Functional inverse of Student's t distribution | | yv | Bessel function of noninteger order | | tandg | Circular tangent of angle in degrees | | cotdg | Circular cotangent of argument in degrees | | log1p | log(1 + x) | | log1pmx | log(1 + x) - x | | expm1 | exp(x) - 1 | | cosm1 | cos(x) - 1 | | lgam1p | lgam(x + 1) | | yn | Bessel function of second kind of integer order | | zeta | Riemann zeta function of two arguments | | zetac | Riemann zeta function | | owens-t | Owen's T-Function |

Contributing

When contributing to this repository, please first discuss major changes to the existing code you wish to make via a github issue. Minor changes and major additions are welcome. Please write good commit messages.

License

CEPHES.CL is available under the Microsoft Public License.


2 Systems

The main system appears first, followed by any subsystem dependency.


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2.1 cephes

Wrapper for the Cephes Mathematical Library

Author

Steven Nunez <steve@symbolics.tech>

License

MS-PL

Version

1.2

Dependency

cffi (system).

Source

cephes.asd.

Child Components

3 Modules

Modules are listed depth-first from the system components tree.


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3.1 cephes/libmd

Source

cephes.asd.

Parent Component

cephes (system).

Child Component

makefile.m (file).


4 Files

Files are sorted by type and then listed depth-first from the systems components trees.


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4.1 Lisp


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4.1.1 cephes/cephes.asd

Source

cephes.asd.

Parent Component

cephes (system).

ASDF Systems

cephes.


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4.1.2 cephes/package.lisp

Dependency

libmd (module).

Source

cephes.asd.

Parent Component

cephes (system).

Packages

cephes.


4.1.3 cephes/init.lisp

Dependency

package.lisp (file).

Source

cephes.asd.

Parent Component

cephes (system).


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4.1.4 cephes/cephes.lisp

Dependency

init.lisp (file).

Source

cephes.asd.

Parent Component

cephes (system).

Public Interface
Internals

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4.2 Source


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4.2.1 cephes/libmd/makefile.m

Source

cephes.asd.

Parent Component

libmd (module).


5 Packages

Packages are listed by definition order.


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5.1 cephes

Source

package.lisp.

Use List

common-lisp.

Public Interface
Internals

6 Definitions

Definitions are sorted by export status, category, package, and then by lexicographic order.


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6.1 Public Interface


6.1.1 Ordinary functions

Function: airy (x)

Solution of the differential equation y”(x) = xy
The function returns the two independent solutions Ai, Bi and their first derivatives Ai’(x), Bi’(x), as VALUES (Ai Bi Aip Bip)

Package

cephes.

Source

cephes.lisp.

Function: bdtr (k n p)

Returns the sum of the terms 0 through k of the Binomial probability density

Package

cephes.

Source

cephes.lisp.

Function: bdtrc (k n p)

Returns the sum of the terms k+1 through n of the Binomial probability density

Package

cephes.

Source

cephes.lisp.

Function: bdtri (k n y)

Finds the event probability p such that the sum of the terms 0 through k of the Binomial probability densityis equal to the given cumulative probability y.

Package

cephes.

Source

cephes.lisp.

Function: besselpoly (a lambda nu)
Package

cephes.

Source

cephes.lisp.

Function: beta (a b)
Package

cephes.

Source

cephes.lisp.

Function: btdtr (a b x)

Returns the area from zero to x under the beta density function.

x
- -
| (a+b) | | a-1 b-1
P(x) = ———- | t (1-t) dt
- - | |
| (a) | (b) -
0

This function is identical to the incomplete beta integral function incbet(a, b, x).

Package

cephes.

Source

cephes.lisp.

Function: cbrt (x)

Returns the cube root of the argument, which may be negative.

Package

cephes.

Source

cephes.lisp.

Function: chdtr (df x)

Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function with DF degrees of freedom.

Package

cephes.

Source

cephes.lisp.

Function: chdtrc (df x)

Returns the area under the right hand tail (from x to infinity) of the Chi square probability density function with DF degrees of freedom

Package

cephes.

Source

cephes.lisp.

Function: chdtri (df y)

Finds the Chi-square argument x such that the integral from x to infinity of the Chi-square density is equal to the given cumulative probability y

Package

cephes.

Source

cephes.lisp.

Function: cosdg (d m s)

Range reduction is into intervals of 45 degrees.

Package

cephes.

Source

cephes.lisp.

Function: cosm1 (x)
Package

cephes.

Source

cephes.lisp.

Function: cospi (x)
Package

cephes.

Source

cephes.lisp.

Function: cotdg (x)

Returns the circular cotangent of the argument x in degrees

Package

cephes.

Source

cephes.lisp.

Function: dawsn (xx)
Package

cephes.

Source

cephes.lisp.

Function: ellie (phi m)
Package

cephes.

Source

cephes.lisp.

Function: ellik (phi m)
Package

cephes.

Source

cephes.lisp.

Function: ellpe (x)
Package

cephes.

Source

cephes.lisp.

Function: ellpk (x)
Package

cephes.

Source

cephes.lisp.

Function: erf (x)
Package

cephes.

Source

cephes.lisp.

Function: erfc (a)
Package

cephes.

Source

cephes.lisp.

Function: erfcinv (y)

Computes the inverse of the complimentary error function on the restricted domain 0 < y < 2. This restriction ensures the existence of a unique result such that erfc(erfcinv(y)) = y.

Package

cephes.

Source

cephes.lisp.

Function: erfinv (y)

Inverse of the error function.
Computes the inverse of the error function on the restricted domain -1 < y < 1. This restriction ensures the existence of a unique result such that erf(erfinv(y)) = y.

Package

cephes.

Source

cephes.lisp.

Function: exp10 (x)

Returns 10 raised to the x power.

Package

cephes.

Source

cephes.lisp.

Function: exp2 (x)

Returns 2 raised to the x power.

Package

cephes.

Source

cephes.lisp.

Function: expm1 (x)
Package

cephes.

Source

cephes.lisp.

Function: expn (n x)

Evaluates the exponential integral

Package

cephes.

Source

cephes.lisp.

Function: fdtr (a b x)

Returns the area from zero to x under the F density function

Package

cephes.

Source

cephes.lisp.

Function: fdtrc (a b x)

Returns the area from x to infinity under the F density function

Package

cephes.

Source

cephes.lisp.

Function: fdtri (a b y)

Finds the F density argument x such that the integral from -infinity to x of the F density is equal to the given probability p

Package

cephes.

Source

cephes.lisp.

Function: fresnl (xxa)
Package

cephes.

Source

cephes.lisp.

Function: gamma (x)

Returns Gamma function of the argument. The result is correctly signed.

Package

cephes.

Source

cephes.lisp.

Function: gdtr (a b x)

Returns the integral from zero to x of the Gamma probability density function

Package

cephes.

Source

cephes.lisp.

Function: gdtrc (a b x)

Returns the integral from x to infinity of the Gamma probability density function

Package

cephes.

Source

cephes.lisp.

Function: gdtri (a b y)
Package

cephes.

Source

cephes.lisp.

Function: hyp2f1 (a b c x)
Package

cephes.

Source

cephes.lisp.

Function: hyperg (a b x)

Computes the confluent hypergeometric function

Package

cephes.

Source

cephes.lisp.

Function: i0 (x)

Returns modified Bessel function of order zero of the argument

Package

cephes.

Source

cephes.lisp.

Function: i0e (x)

Returns exponentially scaled modified Bessel function of order zero of the argument

Package

cephes.

Source

cephes.lisp.

Function: i1 (x)

Returns modified Bessel function of order one of the argument

Package

cephes.

Source

cephes.lisp.

Function: i1e (x)

Returns exponentially scaled modified Bessel function of order one of the argument

Package

cephes.

Source

cephes.lisp.

Function: igam (a x)
Package

cephes.

Source

cephes.lisp.

Function: igamc (a x)
Package

cephes.

Source

cephes.lisp.

Function: igamci (a q)
Package

cephes.

Source

cephes.lisp.

Function: igami (a p)

Returns the x such that: igamc(a, x) = p
The input argument a must be positive and p must be between 0 and 1.

Package

cephes.

Source

cephes.lisp.

Function: incbet (aa bb xx)

Returns incomplete beta integral of the arguments, evaluated from zero to x.

Package

cephes.

Source

cephes.lisp.

Function: incbi (aa bb yy0)

Given y, the function finds x such that incbet( a, b, x ) = y

Package

cephes.

Source

cephes.lisp.

Function: iv (v x)

Returns modified Bessel function of order v of the argument. If x is negative, v must be integer valued.

Package

cephes.

Source

cephes.lisp.

Function: j0 (x)

Returns Bessel function of order zero of the argument

Package

cephes.

Source

cephes.lisp.

Function: j1 (x)

Returns Bessel function of order one of the argument.

Package

cephes.

Source

cephes.lisp.

Function: jacobian-elliptic (u m)

Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), and dn(u|m) of parameter m between 0 and 1, and real argument u. Returns VALUES (sn cn dn)

Package

cephes.

Source

cephes.lisp.

Function: jv (v x)

Returns Bessel function of order v of the argument, where v is real. Negative x is allowed if v is an integer.

Package

cephes.

Source

cephes.lisp.

Function: k0 (x)

Returns modified Bessel function of the third kind of order zero of the argument.

Package

cephes.

Source

cephes.lisp.

Function: k0e (x)

Returns exponentially scaled modified Bessel function of the third kind of order zero of the argument.

Package

cephes.

Source

cephes.lisp.

Function: k1 (x)

Computes the modified Bessel function of the third kind of order one of the argument.

Package

cephes.

Source

cephes.lisp.

Function: k1e (x)

Returns exponentially scaled modified Bessel function of the third kind of order one of the argument

Package

cephes.

Source

cephes.lisp.

Function: kn (nn x)

Returns modified Bessel function of the third kind of order n of the argument

Package

cephes.

Source

cephes.lisp.

Function: lanczos-sum (x)
Package

cephes.

Source

cephes.lisp.

Function: lanczos-sum-near-1 (x)
Package

cephes.

Source

cephes.lisp.

Function: lanczos-sum-near-2 (x)
Package

cephes.

Source

cephes.lisp.

Function: lanczos-sum-scaled (x)
Package

cephes.

Source

cephes.lisp.

Function: lbeta (a b)
Package

cephes.

Source

cephes.lisp.

Function: lgam1p (x)
Package

cephes.

Source

cephes.lisp.

Function: log-gamma (x)

Returns the base e logarithm of the absolute value of the Gamma function of the argument.

Package

cephes.

Source

cephes.lisp.

Function: log-ndtr (a)
Package

cephes.

Source

cephes.lisp.

Function: log1p (x)
Package

cephes.

Source

cephes.lisp.

Function: log1pmx (x)
Package

cephes.

Source

cephes.lisp.

Function: nbdtr (k n p)

Returns the sum of the terms 0 through k of the negative binomial distribution

Package

cephes.

Source

cephes.lisp.

Function: nbdtrc (k n p)

Returns the sum of the terms k+1 to infinity of the negative binomial distribution

Package

cephes.

Source

cephes.lisp.

Function: nbdtri (k n p)

Returns the sum of the terms k+1 to infinity of the negative binomial distribution

Package

cephes.

Source

cephes.lisp.

Function: ndtr (a)

Returns the area under the Gaussian probability density function, integrated from minus infinity to x

Package

cephes.

Source

cephes.lisp.

Function: ndtri (y0)

Returns the argument, x, for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to y.
For small arguments 0 < y < exp(-2), the program computes z = sqrt( -2.0 * log(y) ); then the approximation is x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).

Package

cephes.

Source

cephes.lisp.

Function: owens-t (h a)
Package

cephes.

Source

cephes.lisp.

Function: pdtr (k m)

Returns the sum of the first k terms of the Poisson distribution

Package

cephes.

Source

cephes.lisp.

Function: pdtrc (k m)

Returns the sum of the terms k+1 to infinity of the Poisson distribution

Package

cephes.

Source

cephes.lisp.

Function: pdtri (k y)

Finds the Poisson variable x such that the integral from 0 to x of the Poisson density is equal to the given probability y

Package

cephes.

Source

cephes.lisp.

Function: poch (x m)
Package

cephes.

Source

cephes.lisp.

Function: psi (x)

Returns the logarithmic derivative of the gamma function

Package

cephes.

Source

cephes.lisp.

Function: rgamma (x)

Returns one divided by the Gamma function of the argument

Package

cephes.

Source

cephes.lisp.

Function: shichi (x)

Returns VALUES (si ci)

Package

cephes.

Source

cephes.lisp.

Function: sici (x)

Returns VALUES (si ci)

Package

cephes.

Source

cephes.lisp.

Function: sindg (d m s)

Range reduction is into intervals of 45 degrees.

Package

cephes.

Source

cephes.lisp.

Function: sinpi (x)
Package

cephes.

Source

cephes.lisp.

Function: spence (x)
Package

cephes.

Source

cephes.lisp.

Function: stdtr (k t1)

Computes the integral from minus infinity to t of the Student t distribution with integer k > 0 degrees of freedom

Package

cephes.

Source

cephes.lisp.

Function: stdtri (k p)

Given probability p, finds the argument t such that stdtr(k,t) is equal to p

Package

cephes.

Source

cephes.lisp.

Function: tandg (x)

Returns the circular tangent of the argument x in degrees

Package

cephes.

Source

cephes.lisp.

Function: y0 (x)

Bessel function of the second kind, order zero

Package

cephes.

Source

cephes.lisp.

Function: y1 (x)

Returns Bessel function of the second kind of order one of the argument.

Package

cephes.

Source

cephes.lisp.

Function: yn (n x)

Returns Bessel function of order n, where n is a (possibly negative) integer

Package

cephes.

Source

cephes.lisp.

Function: yv (v x)
Package

cephes.

Source

cephes.lisp.

Function: zeta (x q)
Package

cephes.

Source

cephes.lisp.

Function: zetac (x)
Package

cephes.

Source

cephes.lisp.


6.2 Internals


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6.2.1 Ordinary functions

Function: cephes-airy (x ai aip bi bip)

Solution of the differential equation y”(x) = xy
The function returns the two independent solutions Ai, Bi and their first derivatives Ai’(x), Bi’(x).

Package

cephes.

Source

cephes.lisp.

Function: cephes-ellpj (u m sn cn dn phi)
Package

cephes.

Source

cephes.lisp.

Function: cephes-fresnl (xxa ssa cca)

Evaluates S and C fresnel integrals and returns VALUES (S C)

Package

cephes.

Source

cephes.lisp.

Function: cephes-shichi (x si ci)
Package

cephes.

Source

cephes.lisp.

Function: cephes-sici (x si ci)
Package

cephes.

Source

cephes.lisp.

Function: sign-gamma (x)
Package

cephes.

Source

cephes.lisp.


Appendix A Indexes


Next: , Previous: , Up: Indexes   [Contents][Index]

A.1 Concepts


Next: , Previous: , Up: Indexes   [Contents][Index]

A.2 Functions

Jump to:   A   B   C   D   E   F   G   H   I   J   K   L   N   O   P   R   S   T   Y   Z  
Index Entry  Section

A
airy: Public ordinary functions

B
bdtr: Public ordinary functions
bdtrc: Public ordinary functions
bdtri: Public ordinary functions
besselpoly: Public ordinary functions
beta: Public ordinary functions
btdtr: Public ordinary functions

C
cbrt: Public ordinary functions
cephes-airy: Private ordinary functions
cephes-ellpj: Private ordinary functions
cephes-fresnl: Private ordinary functions
cephes-shichi: Private ordinary functions
cephes-sici: Private ordinary functions
chdtr: Public ordinary functions
chdtrc: Public ordinary functions
chdtri: Public ordinary functions
cosdg: Public ordinary functions
cosm1: Public ordinary functions
cospi: Public ordinary functions
cotdg: Public ordinary functions

D
dawsn: Public ordinary functions

E
ellie: Public ordinary functions
ellik: Public ordinary functions
ellpe: Public ordinary functions
ellpk: Public ordinary functions
erf: Public ordinary functions
erfc: Public ordinary functions
erfcinv: Public ordinary functions
erfinv: Public ordinary functions
exp10: Public ordinary functions
exp2: Public ordinary functions
expm1: Public ordinary functions
expn: Public ordinary functions

F
fdtr: Public ordinary functions
fdtrc: Public ordinary functions
fdtri: Public ordinary functions
fresnl: Public ordinary functions
Function, airy: Public ordinary functions
Function, bdtr: Public ordinary functions
Function, bdtrc: Public ordinary functions
Function, bdtri: Public ordinary functions
Function, besselpoly: Public ordinary functions
Function, beta: Public ordinary functions
Function, btdtr: Public ordinary functions
Function, cbrt: Public ordinary functions
Function, cephes-airy: Private ordinary functions
Function, cephes-ellpj: Private ordinary functions
Function, cephes-fresnl: Private ordinary functions
Function, cephes-shichi: Private ordinary functions
Function, cephes-sici: Private ordinary functions
Function, chdtr: Public ordinary functions
Function, chdtrc: Public ordinary functions
Function, chdtri: Public ordinary functions
Function, cosdg: Public ordinary functions
Function, cosm1: Public ordinary functions
Function, cospi: Public ordinary functions
Function, cotdg: Public ordinary functions
Function, dawsn: Public ordinary functions
Function, ellie: Public ordinary functions
Function, ellik: Public ordinary functions
Function, ellpe: Public ordinary functions
Function, ellpk: Public ordinary functions
Function, erf: Public ordinary functions
Function, erfc: Public ordinary functions
Function, erfcinv: Public ordinary functions
Function, erfinv: Public ordinary functions
Function, exp10: Public ordinary functions
Function, exp2: Public ordinary functions
Function, expm1: Public ordinary functions
Function, expn: Public ordinary functions
Function, fdtr: Public ordinary functions
Function, fdtrc: Public ordinary functions
Function, fdtri: Public ordinary functions
Function, fresnl: Public ordinary functions
Function, gamma: Public ordinary functions
Function, gdtr: Public ordinary functions
Function, gdtrc: Public ordinary functions
Function, gdtri: Public ordinary functions
Function, hyp2f1: Public ordinary functions
Function, hyperg: Public ordinary functions
Function, i0: Public ordinary functions
Function, i0e: Public ordinary functions
Function, i1: Public ordinary functions
Function, i1e: Public ordinary functions
Function, igam: Public ordinary functions
Function, igamc: Public ordinary functions
Function, igamci: Public ordinary functions
Function, igami: Public ordinary functions
Function, incbet: Public ordinary functions
Function, incbi: Public ordinary functions
Function, iv: Public ordinary functions
Function, j0: Public ordinary functions
Function, j1: Public ordinary functions
Function, jacobian-elliptic: Public ordinary functions
Function, jv: Public ordinary functions
Function, k0: Public ordinary functions
Function, k0e: Public ordinary functions
Function, k1: Public ordinary functions
Function, k1e: Public ordinary functions
Function, kn: Public ordinary functions
Function, lanczos-sum: Public ordinary functions
Function, lanczos-sum-near-1: Public ordinary functions
Function, lanczos-sum-near-2: Public ordinary functions
Function, lanczos-sum-scaled: Public ordinary functions
Function, lbeta: Public ordinary functions
Function, lgam1p: Public ordinary functions
Function, log-gamma: Public ordinary functions
Function, log-ndtr: Public ordinary functions
Function, log1p: Public ordinary functions
Function, log1pmx: Public ordinary functions
Function, nbdtr: Public ordinary functions
Function, nbdtrc: Public ordinary functions
Function, nbdtri: Public ordinary functions
Function, ndtr: Public ordinary functions
Function, ndtri: Public ordinary functions
Function, owens-t: Public ordinary functions
Function, pdtr: Public ordinary functions
Function, pdtrc: Public ordinary functions
Function, pdtri: Public ordinary functions
Function, poch: Public ordinary functions
Function, psi: Public ordinary functions
Function, rgamma: Public ordinary functions
Function, shichi: Public ordinary functions
Function, sici: Public ordinary functions
Function, sign-gamma: Private ordinary functions
Function, sindg: Public ordinary functions
Function, sinpi: Public ordinary functions
Function, spence: Public ordinary functions
Function, stdtr: Public ordinary functions
Function, stdtri: Public ordinary functions
Function, tandg: Public ordinary functions
Function, y0: Public ordinary functions
Function, y1: Public ordinary functions
Function, yn: Public ordinary functions
Function, yv: Public ordinary functions
Function, zeta: Public ordinary functions
Function, zetac: Public ordinary functions

G
gamma: Public ordinary functions
gdtr: Public ordinary functions
gdtrc: Public ordinary functions
gdtri: Public ordinary functions

H
hyp2f1: Public ordinary functions
hyperg: Public ordinary functions

I
i0: Public ordinary functions
i0e: Public ordinary functions
i1: Public ordinary functions
i1e: Public ordinary functions
igam: Public ordinary functions
igamc: Public ordinary functions
igamci: Public ordinary functions
igami: Public ordinary functions
incbet: Public ordinary functions
incbi: Public ordinary functions
iv: Public ordinary functions

J
j0: Public ordinary functions
j1: Public ordinary functions
jacobian-elliptic: Public ordinary functions
jv: Public ordinary functions

K
k0: Public ordinary functions
k0e: Public ordinary functions
k1: Public ordinary functions
k1e: Public ordinary functions
kn: Public ordinary functions

L
lanczos-sum: Public ordinary functions
lanczos-sum-near-1: Public ordinary functions
lanczos-sum-near-2: Public ordinary functions
lanczos-sum-scaled: Public ordinary functions
lbeta: Public ordinary functions
lgam1p: Public ordinary functions
log-gamma: Public ordinary functions
log-ndtr: Public ordinary functions
log1p: Public ordinary functions
log1pmx: Public ordinary functions

N
nbdtr: Public ordinary functions
nbdtrc: Public ordinary functions
nbdtri: Public ordinary functions
ndtr: Public ordinary functions
ndtri: Public ordinary functions

O
owens-t: Public ordinary functions

P
pdtr: Public ordinary functions
pdtrc: Public ordinary functions
pdtri: Public ordinary functions
poch: Public ordinary functions
psi: Public ordinary functions

R
rgamma: Public ordinary functions

S
shichi: Public ordinary functions
sici: Public ordinary functions
sign-gamma: Private ordinary functions
sindg: Public ordinary functions
sinpi: Public ordinary functions
spence: Public ordinary functions
stdtr: Public ordinary functions
stdtri: Public ordinary functions

T
tandg: Public ordinary functions

Y
y0: Public ordinary functions
y1: Public ordinary functions
yn: Public ordinary functions
yv: Public ordinary functions

Z
zeta: Public ordinary functions
zetac: Public ordinary functions

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A.3 Variables