### Highlights fromSpecial 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
• eulergamma Euler-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 | 18 ratings Rate this file 23 Downloads (last 30 days) File Size: 37.1 KB File ID: #978

# Special Functions math library

23 Oct 2001 (Updated 26 Aug 2004)

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)
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24 Jul 2012

I mean Paul, sorry.

24 Jul 2012

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

28 Jun 2012

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)

10 Jan 2012

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

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

18 Sep 2011

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

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
19 Oct 2005

This is exactly wat I needed, works perfect

18 May 2005

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

13 Jan 2005
06 Sep 2004
07 Aug 2004

Very useful!

07 Jun 2004

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

14 Mar 2004

02 Dec 2003
08 Sep 2003

Very Useful.
Needs a similar routine for
polygamma.

15 Aug 2003

Thanks, I found it very useful!

08 Jul 2003

11 Jun 2003

very useful

05 Jan 2003

very good

05 Jan 2003

very good

26 Aug 2004

Update to my earlier submissions. Adds an arbitrary order complex polygamma function, pochhammer, harmonic sum. Includes complex gamma, loggamma, bernoulli, euler numbers, Riemann zeta, factorial, binomial, and other functions.