Steepest descent of integrand with a movable saddle?

Question

I want to apply the steepest descent method to the following integration: $$ \int_0^\infty e^{-x^2 + i \sqrt{x^2 + 1} \cdot \lambda } dx $$

It has movable saddle so I need to transform it into the standard form, something like

$$ \int_C g(z) e^{\lambda f(z) } dz $$

I know for Gamma function: $$ \Gamma(x+1)= \int_0^\infty e^{-t} t^{x} dt =\int_0^\infty e^{-t + x \ln t} dt $$ letting $t = x s$ transforms it into standard form. And for Airy function:

$$Ai(x) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i (t^3/3 + x t)} dt $$

letting $t = \sqrt{x} z$ does the trick.

However, no change of variable seems to transform my integration into the standard form. For this reason I can not proceed at all. Any hint or suggestion ? Thanks!

Answer 1

Let us here sketched a derivation.

1. OP's integral has a saddle point at $z=0$ with angular steepest descent direction $\frac{\pi}{4}$. Therefore, inspired by the method of steepest descent, we rewrite OP's integral as $$I(\lambda) ~:=~\int_{\mathbb{R}_+}\!\mathrm{d}z~ \exp\left\{-z^2+i\lambda\sqrt{1+z^2}\right\} ~=~ \frac{e^{i\lambda}}{2}J(\lambda),\qquad \lambda~>~0, \tag{1}$$ where $$\begin{align}J(\lambda) ~:=~&\int_{\mathbb{R}}\!\mathrm{d}z~ \exp\left\{-z^2+i\lambda\left(\sqrt{1+z^2}-1\right)\right\} \cr ~\stackrel{z=\frac{u}{\sqrt{\lambda}}\exp\left(\frac{i\pi}{4}\right)}{=}& \frac{1}{\sqrt{\lambda}}\exp\left(\frac{i\pi}{4}\right) K(\lambda),\end{align}\tag{2} $$ where $$ K(\lambda)~:=~\int_{\mathbb{R}}\!\mathrm{d}u~ \exp\left\{-\frac{iu^2}{\lambda}-f_{\lambda}(u)\right\},\tag{3}$$ and where $$ -f_{\lambda}(u)~:=~i\lambda\left(\sqrt{1+\frac{iu^2}{\lambda}}-1\right) ~=~-\frac{u^2}{2} + \frac{iu^4}{8\lambda}+ O(u^6\lambda^{-2}).\tag{4} $$

2. Let us define the non-negative function $$ f(u)~:=~\frac{u^2}{2}. \tag{5} $$ Clearly $$ f_{\lambda}(u)~\longrightarrow~ f(u)\quad\text{for}\quad\lambda~\to~ \infty \tag{6}$$ $u$-pointwise.

3. We note that the real part is given by $$ {\rm Re}~ f_{\lambda}(u)~=~\lambda\sqrt{\frac{\sqrt{1+\frac{u^4}{\lambda^2}}-1}{2}} .\tag{7}$$ Define a non-negative function $$ g(u)~:=~ \frac{|u|}{2}~\theta(|u|\!-\!\sqrt{2}) , \tag{8}$$ where $\theta$ denotes the Heaviside step function. One may show that $$\forall\lambda\geq\frac{1}{\sqrt{2}}\forall u\in\mathbb{R}:~~ {\rm Re}~ f_{\lambda}(u) ~\geq~ g(u),\tag{9}$$ because $$\forall\lambda\geq\frac{1}{\sqrt{2}}\forall u\geq\sqrt{2}:~~ \sqrt{1+\frac{u^4}{\lambda^2}}-1 ~\geq~ \frac{u^2}{2\lambda^2}.\tag{10}$$

4. It follows that $\exp\left\{-g(u)\right\}$ is a majorant function for the integrand (3). Lebesgue's dominated convergence theorem then shows that the corresponding integral (3) satisfies $$ K(\lambda)~\longrightarrow~ \int_{\mathbb{R}}\!\mathrm{d}u~ \exp\left\{-f(u)\right\}~=~\sqrt{2\pi} \quad \text{for} \quad\lambda~\to~ \infty. \tag{11} $$

5. Next we can estimate corrections to any order in $1/\lambda$ we like. E.g. $$\begin{align}K(\lambda)~\stackrel{(3)+(4)}{=}&\int_{\mathbb{R}}\!\mathrm{d}u~ \exp\left\{-\left(\frac{i}{\lambda}+\frac{1}{2}\right)u^2\right\} \left(1 +\frac{iu^4}{8\lambda} + O(\lambda^{-2})\right)\cr ~=~&\sqrt{\frac{\pi}{\frac{i}{\lambda}+\frac{1}{2}}}+\frac{i}{8\lambda}\underbrace{\int_{\mathbb{R}}\!\mathrm{d}u~\exp\left\{-\frac{u^2}{2}\right\}u^4}_{=3\sqrt{2\pi}} + O(\lambda^{-2})\cr ~=~&\sqrt{2\pi}\left(1 -\frac{5i}{8\lambda} + O(\lambda^{-2})\right) ,\end{align}\tag{12}$$ and so forth. Hence OP's integral (1) has an asymptotic expansion

   $$ I(\lambda)~\sim~ e^{i\lambda}\frac{1+i}{2}\sqrt{\frac{\pi}{\lambda}}\left(1 -\frac{5i}{8\lambda} + O(\lambda^{-2})\right)\quad \text{for}\quad \lambda~\to~\infty .\tag{13} $$

Answer 2

Not a full answer but too long for a comment.

It appears that the saddle point at $x=0$ is dominant. There are two more but they can be avoided (and they possibly lie on the branch cuts of the integrand anyway---that would be bad).

If we make the substitution $x = u/\sqrt{\lambda}$ we get

$$ \begin{align} \int_0^\infty \exp\left\{-x^2+i\lambda\sqrt{x^2+1}\right\}dx &= \frac{1}{\sqrt{\lambda}} \int_0^\infty \exp\left\{-\frac{u^2}{\lambda}+i\lambda\sqrt{\frac{u^2}{\lambda}+1}\right\}du \\ &= \frac{e^{i\lambda}}{\sqrt{\lambda}} \int_0^\infty \exp\left\{-\frac{u^2}{\lambda}-i\lambda \left( 1-\sqrt{\frac{u^2}{\lambda}+1} \right) \right\}du. \end{align} $$

Numerically,

$$ \int_0^\infty \exp\left\{-\frac{u^2}{\lambda}-i\lambda \left( 1-\sqrt{\frac{u^2}{\lambda}+1} \right) \right\}du \to \int_0^\infty \exp\{iu^2/2\}\,du = \frac{1+i}{2}\sqrt{\pi} $$

as $\lambda \to \infty$, but I'm not sure how to show it at the moment. This would imply that

$$ \int_0^\infty \exp\left\{-x^2+i\lambda\sqrt{x^2+1}\right\}dx \sim \frac{(1+i)e^{i\lambda}}{2} \sqrt{\frac{\pi}{\lambda}} $$

as $\lambda \to \infty$.