The Cephes Reference Manual

Table of Contents

The Cephes Reference Manual

This is the Cephes Reference Manual, version 1.1.

1 Systems

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

1.1 cephes

Author

Steven Nunez <steve@symbolics.tech>

License

MS-PL

Description

Wrapper for the Cephes Mathematical Library

Version

1.1

Dependency

cffi

Source

cephes.asd (file)

Directory

s:/src/cephes/

Components

• package.lisp (file)
• init.lisp (file)
• cephes.lisp (file)

2 Files

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

2.1 Lisp

2.1.1 cephes.asd

Location

cephes.asd

Systems

cephes (system)

2.1.2 cephes/package.lisp

Parent

cephes (system)

Location

package.lisp

Packages

cephes

2.1.3 cephes/init.lisp

Dependency

package.lisp (file)

Parent

cephes (system)

Location

init.lisp

2.1.4 cephes/cephes.lisp

Dependency

init.lisp (file)

Parent

cephes (system)

Location

cephes.lisp

Exported Definitions

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

Internal Definitions

• cephes-airy (function)
• cephes-ellpj (function)
• cephes-fresnl (function)
• cephes-shichi (function)
• cephes-sici (function)
• sign-gamma (function)

3 Packages

Packages are listed by definition order.

3.1 cephes

Source

package.lisp (file)

Use List

common-lisp

Exported Definitions

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

Internal Definitions

• cephes-airy (function)
• cephes-ellpj (function)
• cephes-fresnl (function)
• cephes-shichi (function)
• cephes-sici (function)
• sign-gamma (function)

4 Definitions

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

4.1 Exported definitions

4.1.1 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 (file)

Function: bdtr K N P

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

Package

cephes

Source

cephes.lisp (file)

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 (file)

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 (file)

Function: besselpoly A LAMBDA NU

Package

cephes

Source

cephes.lisp (file)

Function: beta A B

Package

cephes

Source

cephes.lisp (file)

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 (file)

Function: cbrt X

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

Package

cephes

Source

cephes.lisp (file)

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 (file)

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 (file)

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 (file)

Function: cosdg D M S

Range reduction is into intervals of 45 degrees.

Package

cephes

Source

cephes.lisp (file)

Function: cosm1 X

Package

cephes

Source

cephes.lisp (file)

Function: cospi X

Package

cephes

Source

cephes.lisp (file)

Function: cotdg X

Returns the circular cotangent of the argument x in degrees

Package

cephes

Source

cephes.lisp (file)

Function: dawsn XX

Package

cephes

Source

cephes.lisp (file)

Function: ellie PHI M

Package

cephes

Source

cephes.lisp (file)

Function: ellik PHI M

Package

cephes

Source

cephes.lisp (file)

Function: ellpe X

Package

cephes

Source

cephes.lisp (file)

Function: ellpk X

Package

cephes

Source

cephes.lisp (file)

Function: erf X

Package

cephes

Source

cephes.lisp (file)

Function: erfc A

Package

cephes

Source

cephes.lisp (file)

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 (file)

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 (file)

Function: exp10 X

Returns 10 raised to the x power.

Package

cephes

Source

cephes.lisp (file)

Function: exp2 X

Returns 2 raised to the x power.

Package

cephes

Source

cephes.lisp (file)

Function: expm1 X

Package

cephes

Source

cephes.lisp (file)

Function: expn N X

Evaluates the exponential integral

Package

cephes

Source

cephes.lisp (file)

Function: fdtr A B X

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

Package

cephes

Source

cephes.lisp (file)

Function: fdtrc A B X

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

Package

cephes

Source

cephes.lisp (file)

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 (file)

Function: fresnl XXA

Package

cephes

Source

cephes.lisp (file)

Function: gamma X

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

Package

cephes

Source

cephes.lisp (file)

Function: gdtr A B X

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

Package

cephes

Source

cephes.lisp (file)

Function: gdtrc A B X

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

Package

cephes

Source

cephes.lisp (file)

Function: gdtri A B Y

Package

cephes

Source

cephes.lisp (file)

Function: hyp2f1 A B C X

Package

cephes

Source

cephes.lisp (file)

Function: hyperg A B X

Computes the confluent hypergeometric function

Package

cephes

Source

cephes.lisp (file)

Function: i0 X

Returns modified Bessel function of order zero of the argument

Package

cephes

Source

cephes.lisp (file)

Function: i0e X

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

Package

cephes

Source

cephes.lisp (file)

Function: i1 X

Returns modified Bessel function of order one of the argument

Package

cephes

Source

cephes.lisp (file)

Function: i1e X

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

Package

cephes

Source

cephes.lisp (file)

Function: igam A X

Package

cephes

Source

cephes.lisp (file)

Function: igamc A X

Package

cephes

Source

cephes.lisp (file)

Function: igamci A Q

Package

cephes

Source

cephes.lisp (file)

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 (file)

Function: incbet AA BB XX

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

Package

cephes

Source

cephes.lisp (file)

Function: incbi AA BB YY0

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

Package

cephes

Source

cephes.lisp (file)

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 (file)

Function: j0 X

Returns Bessel function of order zero of the argument

Package

cephes

Source

cephes.lisp (file)

Function: j1 X

Returns Bessel function of order one of the argument.

Package

cephes

Source

cephes.lisp (file)

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 (file)

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 (file)

Function: k0 X

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

Package

cephes

Source

cephes.lisp (file)

Function: k0e X

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

Package

cephes

Source

cephes.lisp (file)

Function: k1 X

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

Package

cephes

Source

cephes.lisp (file)

Function: k1e X

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

Package

cephes

Source

cephes.lisp (file)

Function: kn NN X

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

Package

cephes

Source

cephes.lisp (file)

Function: lanczos-sum X

Package

cephes

Source

cephes.lisp (file)

Function: lanczos-sum-near-1 X

Package

cephes

Source

cephes.lisp (file)

Function: lanczos-sum-near-2 X

Package

cephes

Source

cephes.lisp (file)

Function: lanczos-sum-scaled X

Package

cephes

Source

cephes.lisp (file)

Function: lbeta A B

Package

cephes

Source

cephes.lisp (file)

Function: lgam1p X

Package

cephes

Source

cephes.lisp (file)

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 (file)

Function: log-ndtr A

Package

cephes

Source

cephes.lisp (file)

Function: log1p X

Package

cephes

Source

cephes.lisp (file)

Function: log1pmx X

Package

cephes

Source

cephes.lisp (file)

Function: nbdtr K N P

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

Package

cephes

Source

cephes.lisp (file)

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 (file)

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 (file)

Function: ndtr A

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

Package

cephes

Source

cephes.lisp (file)

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 (file)

Function: owens-t H A

Package

cephes

Source

cephes.lisp (file)

Function: pdtr K M

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

Package

cephes

Source

cephes.lisp (file)

Function: pdtrc K M

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

Package

cephes

Source

cephes.lisp (file)

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 (file)

Function: poch X M

Package

cephes

Source

cephes.lisp (file)

Function: psi X

Returns the logarithmic derivative of the gamma function

Package

cephes

Source

cephes.lisp (file)

Function: rgamma X

Returns one divided by the Gamma function of the argument

Package

cephes

Source

cephes.lisp (file)

Function: shichi X

Returns VALUES (si ci)

Package

cephes

Source

cephes.lisp (file)

Function: sici X

Returns VALUES (si ci)

Package

cephes

Source

cephes.lisp (file)

Function: sindg D M S

Range reduction is into intervals of 45 degrees.

Package

cephes

Source

cephes.lisp (file)

Function: sinpi X

Package

cephes

Source

cephes.lisp (file)

Function: spence X

Package

cephes

Source

cephes.lisp (file)

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 (file)

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 (file)

Function: tandg X

Returns the circular tangent of the argument x in degrees

Package

cephes

Source

cephes.lisp (file)

Function: y0 X

Bessel function of the second kind, order zero

Package

cephes

Source

cephes.lisp (file)

Function: y1 X

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

Package

cephes

Source

cephes.lisp (file)

Function: yn N X

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

Package

cephes

Source

cephes.lisp (file)

Function: yv V X

Package

cephes

Source

cephes.lisp (file)

Function: zeta X Q

Package

cephes

Source

cephes.lisp (file)

Function: zetac X

Package

cephes

Source

cephes.lisp (file)

4.2 Internal definitions

4.2.1 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 (file)

Function: cephes-ellpj U M SN CN DN PHI

Package

cephes

Source

cephes.lisp (file)

Function: cephes-fresnl XXA SSA CCA

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

Package

cephes

Source

cephes.lisp (file)

Function: cephes-shichi X SI CI

Package

cephes

Source

cephes.lisp (file)

Function: cephes-sici X SI CI

Package

cephes

Source

cephes.lisp (file)

Function: sign-gamma X

Package

cephes

Source

cephes.lisp (file)

Appendix A Indexes

A.1 Concepts

A.2 Functions

A.3 Variables

A.4 Data types