lerch: 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.

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).

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.

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.

Examples

Run this code

# 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")
# }

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