## Integral representation of the modified Bessel functions of the second kind and asymptotic expansion

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The modified Bessel function (Macdonald function) $K_\alpha(z)$ is known to have the following asymptotic expansion for large positive $z$: $$K_\alpha(z)=\sqrt{\frac{\pi}{2z}}e^{-z}\sum_{k=0}^\infty \frac{b_k(\alpha)}{z^ k}$$ where $b_1(\alpha)=1$, $b_2(\alpha)=\frac{\alpha^2-1^2}{1!8}$, $b_3(\alpha)=\frac{(\alpha^2-1^2)(\alpha^2-3^2)}{2!(8)^2}$ and so on. Is there any simple integral representation, for which it would be a perturbative expansion such that $$K_\alpha(z)=h(z) \int_C \exp\left(\frac{f(y)}{z}\right) g(y)^\alpha d\mu(y)$$ where $f(x)$, $g(x)$, $h(x)$ and $d \mu(x)$ are $\alpha$-independent?

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I think you want the integrand to have the factor $\exp(z f(y))$, since you are taking the $z\to\infty$ limit and want the argument of the exponential to vary quickly with $y$. – Igor Khavkine Jan 31 at 15:38

## 1 Answer

The DLMF lists multiple integral representations of $K_\nu(z)$. Here's one that fits your bill:

```
\[\mathop{K_{{\nu}}}\nolimits\!\left(z\right)=\frac{\pi^{{\frac{1}{2}}}(\frac{1}{2}z)^{\nu}}{\mathop{\Gamma}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int _{0}^{\infty}e^{{-z\mathop{\cosh}\nolimits t}}(\mathop{\sinh}\nolimits t)^{{2\nu}}dt .\]```

For integer $\nu$, the contour could be extended by symmetry to all of $\mathbb{R}$.

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Thank you, I know this nice integral representation, but I want something (maybe) more complicated - with $\frac{1}{z}$ as I wrote.  – Sasha Jan 31 at 15:54
Sorry, I don't follow your logic about $1/z$ in $g(f(y)/z)$. What kind of expansion would you expect to perform? The integral formula I suggested should be amenable to a Laplace-type (or steepest descent) expansion for large $z$. In any case, most known integral representations are in fact listed on the DLMF page. – Igor Khavkine Jan 31 at 20:53
I just want to have simple $1/z$ expansion for large $z$ (and not a usual steepest descent expansion). – Sasha Feb 1 at 13:06
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