lerch: Lerch Phi Function

In VGAM: Vector Generalized Linear and Additive Models

Lerch Phi Function

Description

Computes the Lerch Phi function.

Usage

lerch(x, s, v, tolerance = 1.0e-10, iter = 100)

Arguments

x, s, v	Numeric. This function recyles values of x, s, and v if necessary.
tolerance	Numeric. Accuracy required, must be positive and less than 0.01.
iter	Maximum number of iterations allowed to obtain convergence. If iter is too small then a result of NA may occur; if so, try increasing its value.

Details

Also known as the Lerch transcendent, it can be defined by an integral involving analytical continuation. An alternative definition is the series

\Phi(x,s,v) = \sum_{n=0}^{\infty} \frac{x^n}{(n+v)^s}

which converges for |x|<1 as well as for |x|=1 with s>1. The series is undefined for integers v <= 0. Actually, x may be complex but this function only works for real x. The algorithm used is based on the relation

\Phi(x,s,v) = x^m \Phi(x,s,v+m) + \sum_{n=0}^{m-1} \frac{x^n}{(n+v)^s} .

See the URL below for more information. This function is a wrapper function for the C code described below.

Value

Returns the value of the function evaluated at the values of x, s, v. If the above ranges of x and v are not satisfied, or some numeric problems occur, then this function will return an NA for those values. (The C code returns 6 possible return codes, but this is not passed back up to the R level.)

Warning

This function has not been thoroughly tested and contains limitations, for example, the zeta function cannot be computed with this function even though \zeta(s) = \Phi(x=1,s,v=1). Several numerical problems can arise, such as lack of convergence, overflow and underflow, especially near singularities. If any problems occur then an NA will be returned. For example, if |x|=1 and s>1 then convergence may be so slow that changing tolerance and/or iter may be needed to get an answer (that is treated cautiously).

Note

There are a number of special cases, e.g., the Riemann zeta-function is \zeta(s) = \Phi(x=1,s,v=1). Another example is the Hurwitz zeta function \zeta(s, v) = \Phi(x=1,s,v=v). The special case of s=1 corresponds to the hypergeometric 2F1, and this is implemented in the gsl package. The Lerch Phi function should not be confused with the Lerch zeta function though they are quite similar.

Author(s)

S. V. Aksenov and U. D. Jentschura wrote the C code (called Version 1.00). The R wrapper function was written by T. Yee.

References

Originally the code was found at http://aksenov.freeshell.org/lerchphi/source/lerchphi.c.

Bateman, H. (1953). Higher Transcendental Functions. Volume 1. McGraw-Hill, NY, USA.

See Also

zeta.

Examples

## Not run: 
s <- 2; v <- 1; x <- seq(-1.1, 1.1, length = 201)
plot(x, lerch(x, s = s, v = v), type = "l", col = "blue",
     las = 1, main = paste0("lerch(x, s = ", s,", v = ", v, ")"))
abline(v = 0, h = 1, lty = "dashed", col = "gray")

## End(Not run)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.