### Highlights from Special Functions math library

• bern(n)Bern Bernoulli number
• binomial(n,d)BINOMIAL calculate the binomial coefficient
• deta(z,k)DETA Calculates Dirichlet functions of the form
• erfz(zz)ERFZ Error function for complex inputs
• eta(z)ETA Dirichlet Eta function
• euler(n)Euler Euler number
• eulergammaEuler-Mascheroni constant = -Psi(1) = 0.5772156649015328606...
• fact(n)FACT Vectorized Factorial function
• factd(n)FACTD Double Factorial function = n!!
• gamma(z)GAMMA Gamma function valid in the entire complex plane.
• gammaln(z)GAMMALOG Natural Log of the Gamma function valid in the entire complex plane.
• genocchi(z)Genocchi number
• harm(z)Harm Harmonic sum function valid in the entire (complex) plane.
• lambda(z)LAMBDA Dirichlet Lambda function
• poch(z,n)
• psi(z)Psi Psi (or Digamma) function valid in the entire complex plane.
• psin(n,z)Psin Arbitrary order Polygamma function valid in the entire complex plane.
• totient(n)TOTIENT calculates the totient function (also
• zeta(z)ZETA Riemann Zeta function
• View all files
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4.6 | 19 ratings Rate this file 28 Downloads (last 30 days) File Size: 37.1 KB File ID: #978 Version: 1.0

# Special Functions math library

### Paul Godfrey (view profile)

23 Oct 2001 (Updated )

Collection of Special Functions programs.

File Information
Description

A collection of special function programs valid in the entire complex plane. Includes Gamma, loggamma, psi, polygamma, error, zeta, and others.

Editor's Note: Please note that these files have the same names as files already included with MATLAB. Being aware of where they are on your path will help you determine when you're using these files and not the MathWorks versions.

Programs to calculate the complex Gamma, complex LogGamma, complex error, complex psi, complex Riemann zeta, vectorized factorial, vectorized double factorial functions as well as Bernoulli, Euler, Genocchi, and totient numbers.

Acknowledgements

This file inspired Generation Of Random Variates.

MATLAB release MATLAB 6.0 (R12)
14 Jun 2015 Robert Gragg

### Robert Gragg (view profile)

Excellent tools.
MATLAB should have this complex-plane functionality for its built-in erf, gamma, gammaln, and psi.

24 Jul 2012 Martin

### Martin (view profile)

I mean Paul, sorry.

Comment only
24 Jul 2012 Martin

### Martin (view profile)

Hey Peter. Does your erfz function still work in version 7.14.0.0739 which i currently have?

Comment only
28 Jun 2012 Aurelien Queffurust

### Aurelien Queffurust (view profile)

The totient function to compute euler phi could be written in one line:
phi = arrayfun(@(x) fix(max(1,x*prod(1 - 1./unique(factor(x))))),n)

Comment only
10 Jan 2012 Scott

### Scott (view profile)

The package is generally excellent.
My only complaint is that the accuracy of the zeta function deterioriates rapidly for |z|>80.
For example, the script returns
0.541674252238781 - 0.348243845696725i
for the 29-th non-trivial zero, which is approximately 0.5+98.8311i.

If you ever have time for an upgrade it would be much appreciated!

04 Jan 2012 Bien

### Bien (view profile)

Thanks! Needed a erfz function and this works well for me. This should be in the standard MATLAB function.

18 Sep 2011 David Young

### David Young (view profile)

Very helpful and useful (rating and comments based only on gamma and gammaln functions). Suggestions:

1. The code to allocate storage:

f = 0.*z; % reserve space in advance

is not needed (indeed is wasteful), as preallocation is only useful ahead of a loop or to set up an array of a specific shape.

2. It might be worth switching to logical indexing rather than linear indexing - this would avoid the use of find() and the reshape at the end.

3. gammaln(z) returns infinities for abs(imag(z)) greater than about 226 and real(z) < 0. This is due to overflow of sin() in the reflection formula, but it is an unnecessary restriction as log(sin(z)) can be computed without overflow over a larger set of values than can sin(z). For example, we can use log(sin(x + iy)) (x and y real) is approximately equal to y + log(0.5i * exp(-1i * x)) for large positive y, and the approximation is good to machine accuracy if y > 18 or thereabouts. (For negative y the approximation is -y + log(-0.5i * exp(1i * x)).) Replacing log(sin()) by a call to a logsin() function that uses these approximations greatly extends the set of valid arguments.

14 Aug 2011 Ben Petschel

### Ben Petschel (view profile)

Nice work, these should be included in the core MATLAB.

Since erf rapidly goes to infinity along the i axis (e.g. erfz(1i*30) = 1i*Inf in floating point), it would be useful to have a function that calculates exp(z^2)*erf(z) or z*exp(z^2)*erfc(z).

24 Jan 2006 andrzej andjez@gazeta.pl
19 Oct 2005 Bart Selins

This is exactly wat I needed, works perfect

18 May 2005 Jaime Ramirez

Really useful, includes functions that are not easy to find somewhere else

13 Jan 2005 Amine Abdallah
06 Sep 2004 Marinos Vouvakis
07 Aug 2004 Joerg Enderlein

Very useful!

07 Jun 2004 David Terr

Looks great! I haven't even tried most of these yet, but I can tell it's a very useful package.

14 Mar 2004 ETOR AMAFU

Comment only
08 Sep 2003 Peter Theodossiou

Very Useful.
Needs a similar routine for
polygamma.

15 Aug 2003 Laurie Mailhe

Thanks, I found it very useful!

08 Jul 2003 teresa pulido

11 Jun 2003 Nikolai Tolich

very useful

05 Jan 2003 asmail KAHDER

very good

05 Jan 2003 asmail KAHDER

very good