MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Dear colleagues, I want to know if there are some results on the bounds of modified Bessel functions $I_\alpha(x)$ and $K_\alpha(x)$? Especially, I need the exponential bounds for them, that is to say if it is possible to get a result like

$|I_\alpha(x)|\leq C exp(b x), x\geq 0$

where $C$ is a constant or a polynomial of $x$, b is a constant, $\alpha$ is a real number. Also the inequality of the same form about $K_\alpha(x)$.

Thank you in advance!

share|cite|improve this question
The Abramowitz-Stegun Handbook has some asymptotics that may help you. – Zsbán Ambrus Dec 31 '10 at 17:31
Maybe this is going the wrong way round, but another possible bound can be obtained by using $I_\nu(z) = (z/2)e^z M(0.5+\nu; 1+2\nu; 2z) / \Gamma(\nu+1)$, where $M$ is Kummer's function. Now, you can invoke Stirling's approximation for $\Gamma$, and one of the many approximations for $M$. – Suvrit Dec 31 '10 at 18:57
up vote 1 down vote accepted

There are some bounds of that form in this paper. See also the first reference at the end of the paper.

share|cite|improve this answer
Thank you very much, I think your comments will help me a lot! – user11936 Dec 31 '10 at 18:43
A user named "lonewolf" has pointed out that the link is now dead. – S. Carnahan Jun 23 '13 at 23:00

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.