At $ab^2\gg 1$, investigate the leading asymptotic of the integral
$$\int\limits_{0}^{\infty }\cos \left ( ax+\frac{2b}{\sqrt{x}} \right )dx$$
I have a general formula that I have used
$$\int e^{if(x)} g(x) dx \approx \sqrt{\frac{2\pi}{|f''(x_0)|}} e^{if(x_0)} g(x_0)$$
And the transformation of the integral I made
$$\int\limits_{0}^{\infty }\cos \left ( ax+\frac{2b}{\sqrt{x}} \right )dx = \text{Re} \int\limits_{0}^{\infty } e^{i(ax + \frac{2b}{\sqrt{x}})}dx$$
where $f''(x_0)$ is the second derivative of $f(x)$ taken at the stationary point. In this case $g(x) = 1$, and $f''(x) = \frac{3b}{2x^{5/2}}$. In the function $f(x) = ax + \frac{2b}{\sqrt{x}}$, the stationary point $x_0$ is where the derivative $f'(x) = a -\frac{b}{x^{3/2}}$ converges to zero. From this we get
$$a – \frac{b}{x_0^{3/2}} = 0\Rightarrow x_0 = \left(\frac{b}{a}\right)^{2/3}.$$
Then the asymptotics of the integral
$$\int\limits_{0}^{\infty }\cos \left ( ax+\frac{2b}{\sqrt{x}} \right )dx \approx \sqrt{\frac{2\pi}{\frac{3b}{2} \left(\frac{b}{a}\right)^{5/3}}} \cos\left(a\left(\frac{b}{a}\right)^{2/3} + \frac{2b}{\sqrt{\left(\frac{b}{a}\right)^{2/3}}}\right)\approx \sqrt{\frac{4\pi a}{3b}} \cos\left(2\sqrt{ab}\right)$$
In past questions related to asymptotics, a good resource (https://dlmf.nist.gov/10.17#E3) was shared with me. Can similar decompositions be applied in this case as well?
Have I found the asymptotics correctly? I would be glad to help
Best Answer
It is better to bring the integral to a standard form by performing the change of variables from $x$ to $t$ via $x = (b/a)^{2/3} t$. Hence $$ \int_0^{ + \infty } {\!\exp \left( {{\rm i}\left( {ax + \frac{{2b}}{{\sqrt x }}} \right)} \right){\rm d}x} = \left( {\frac{b}{a}} \right)^{2/3} \int_0^{ + \infty } {\!\exp \left( {{\rm i}(ab^2)^{1/3} \left( {t + \frac{2}{{\sqrt t }}} \right)} \right){\rm d}t} $$ The stationary point of $t\mapsto t + \frac{2}{{\sqrt t }}$ is at $1$. Thus, by the method of stationary phase, \begin{align*} \left( {\frac{b}{a}} \right)^{2/3} \int_0^{ + \infty } {\!\exp \left( {{\rm i}(ab^2 )^{1/3} \left( {t + \frac{2}{{\sqrt t }}} \right)} \right){\rm d}t} & \sim \left( {\frac{b}{a}} \right)^{2/3} \exp \left( {{\rm i}\left( {3(ab^2)^{1/3} + \frac{\pi }{4}} \right)} \right)\sqrt {\frac{{4\pi }}{{3(ab^2)^{1/3} }}}\\ & = 2\sqrt {\frac{\pi }{3}} \frac{{b^{1/3} }}{{a^{5/6} }}\exp \left( {{\rm i}\left( {3(ab^2 )^{1/3} + \frac{\pi }{4}} \right)} \right) \end{align*} as $ab^2 \to+\infty$. Consequently, $$ \int_0^{ + \infty } {\cos \left( {ax + \frac{{2b}}{{\sqrt x }}} \right){\rm d}x} \sim 2\sqrt {\frac{\pi }{3}} \frac{{b^{1/3} }}{{a^{5/6} }}\cos \left( {3(b^2 a)^{1/3} + \frac{\pi }{4}} \right) $$ as $ab^2 \to+\infty$.