the Confluent Hypergeometric Function of the 1st Kind

Question

I need to do integration

$$\int_0^\infty e^{-z~\text{cosh}(2u)-\frac{1}{y}u^2}~\text{M}\bigg(-\mu,\frac{3}{2},2z~\text{sinh}(u)^2 \bigg)~ \text{sinh}(2u)~\text{sin}\bigg(\frac{\pi u}{y} \bigg)du $$

where $\text{M}(a,b,c)$ represents the Confluent Hypergeometric function of the the 1st kind.

The simulation is implemented using R, and M(a,b,c) is computed by the function 'kummerM' of R's package fAsianOptions. My question is how to define the upper limit for integration, here it is $\infty$ in the formula.

Assuming $ \mu =-0.75,z=0.5$ and $y = 0.15$. If upper=1000, sinh(1000)=Inf; if upper=100, R crashes before obtaining the result. R can produce the results only when upper limit $\in [1,50]$. What is a 'reasonable'choice of upper limit? Although R can produce result when upper limit $\in [1,50]$, the results are quite different. The best upper limit should be the largest one?

Please see the edited post. @Claude Leibovici – Smirk Aug 25 '17 at 09:16 — Smirk, Aug 25 '17 at 09:16

score 0 · Answer 1 · answered Aug 25 '17 at 09:19

0

This is more of a suggestion, not a complete answer.

Have you tried substituting $\operatorname{sinh}(u)=x$? This way you mitigate the effect of the exponential growth of $\operatorname{sinh}$ to a certain degree.

This wikipedia helps you do the necessary computations: https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions#Derivatives You might also need $\operatorname{sinh}(2x)=2\operatorname{cosh}(x)\operatorname{sinh}(x)$.

On the other hand are you even sure this integral converges? $M(a,b,c)$ does grow exponentially in $c$ and then you plug in the exponentially growing function $\sinh$, which makes it even worse. I am not sure if $e^{-\frac{1}{y}u^2}$ can dampen the resulting effect. Here is another wikipedia to underline the exponential growth: https://en.wikipedia.org/wiki/Confluent_hypergeometric_function#Asymptotic_behavior

answered Aug 25 '17 at 09:19

humanStampedist

3,622

Thanks for your comments! I am not good at math. I learnt this formula from a scientific paper and going to use it to compute some results for my paper. @humanStampedist – Smirk Aug 25 '17 at 09:27
Can you give a reference for this paper? – humanStampedist Aug 25 '17 at 09:29
here is the link: https://wwwf.imperial.ac.uk/~ajacquie/index_files/GJH%20-%20SABRMassZero.pdf You may see my question concerning one part of eq(2.5) on page 4. – Smirk Aug 25 '17 at 09:30
There is a typo in your formula. You have written $e^{-z\cos(2u)}$ but the original paper says $e^{-z\cosh(2u)}$. This term could be enough to dampen the exponential growth of $M$. – humanStampedist Aug 25 '17 at 09:35
You are right! Here is a typo. But in my R code, it is cosh.@humanStampedist – Smirk Aug 25 '17 at 09:37
A silly question, $sinh(u)^2$=$sinh(u)*sinh(u)$, right? @humanStampedist – Smirk Aug 25 '17 at 09:44
Yes, that's right. – humanStampedist Aug 25 '17 at 09:45
Thanks! Then, the best choice of the upper limit should be 50? @humanStampedish – Smirk Aug 25 '17 at 09:48
To be honest, I do not know. Maybe you can use the asymptotic formula given in the above wikipedia article to replace $M$ for values above $50$. That way the integrand should simplify a lot. – humanStampedist Aug 25 '17 at 09:52
I appreciate your advice :) – Smirk Aug 25 '17 at 09:54

the Confluent Hypergeometric Function of the 1st Kind

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