While trying to solve a challenging system of dual integral equations resulting from a mixed boundary value problem arising in a fluid mechanical problem, the four following non-trivial convergent improper integrals emerge:
\begin{align}
I_1 (r,t) &= \int_0^\infty \lambda^{\frac{1}{2}}
e^{-\lambda} J_1 (\lambda r) J_{\frac{1}{2}} (\lambda t) \, \mathrm{d}\lambda \, , \\
I_2 (r,t) &= \int_0^\infty \lambda^{\frac{1}{2}}
e^{-\lambda} J_0 (\lambda r) J_{\frac{3}{2}} (\lambda t) \, \mathrm{d} \lambda\, , \\
I_3 (r,t) &= \int_0^\infty \left( \lambda^{\frac{1}{2}}-\lambda^{-\frac{1}{2}} \right)
e^{-\lambda} J_1 (\lambda r) J_{\frac{3}{2}} (\lambda t) \, \mathrm{d}\lambda \, , \\
I_4 (r,t) &= \int_0^\infty \left( \lambda^{\frac{1}{2}}+\lambda^{-\frac{1}{2}} \right)
e^{-\lambda} J_0 (\lambda r) J_{\frac{1}{2}} (\lambda t) \, \mathrm{d}\lambda \, ,
\end{align}
wherein $t$ and $r$ are positive real numbers.
It can be checked that, thanks to the exponential function, these integrals are convergent.
If the integrands do not contain the exponential function, then the evaluation of these integrals is easy and straightforward.
Is there probably a way to evaluate these integrals analytically even as infinite convergent series functions?
Any help or hint is highly appreciated!
Best Answer
I believe these integrals have a simple analytical form. I will demonstrate for $I_1$ and I hope you can see how to do the others similarly.
I write $I_1$ out as originally stated:
$$I_1 = \int_0^{\infty} d\lambda \, \lambda^{1/2} \, e^{-\lambda} \, J_1(\lambda r) \, J_{1/2}(\lambda t)$$
Note that
$$J_{1/2}(\lambda t) = \sqrt{\frac{2}{\pi \lambda t}} \sin{(\lambda t)}$$ $$J_1(\lambda r) = \frac1{i \pi} \int_0^{\pi} d\theta \, \cos{\theta} \, e^{i \lambda r \cos{\theta}} $$
Plugging back into the integral definition of $I_1$ and changing the order of integration, we get
$$I_1 = \frac1{i \pi} \sqrt{\frac{2}{\pi t}} \int_0^{\pi} d\theta \, \cos{\theta} \, \int_0^{\infty} d\lambda \, e^{-\lambda} \, \sin{(\lambda t)} \, e^{i \lambda r \cos{\theta}} $$
Rewriting the sine in exponential form, the integral over $\lambda$ is simple, and we are left with the integral over $\theta$:
$$I_1 = -\frac1{2 \pi} \sqrt{\frac{2}{\pi t}} \int_0^{\pi} d\theta \, \cos{\theta} \, \left [\frac1{1-i t - i r \cos{\theta}} - \frac1{1+i t - i r \cos{\theta}} \right ] $$
Now let's consider the integral
$$\int_0^{\pi} d\theta \, \frac{\cos{\theta}}{a+b \cos{\theta}} $$
where $a$ and $b$ may be complex; in our case $a=1\pm i t$ and $b=-i r$. While there are at least a couple of ways of evaluating this integral, I will demonstrate how it is done using contour integration.
Consider the contour integral
$$-i \oint_C \frac{dz}{z} \, \frac{z^2+1}{b z^2+2 a z+b} $$
where $C$ is the following contour:
The semicircle has unit radius. Note that, because the real integral is only over a half-cycle rather than a full cycle, the contour $C$ includes a traversal along the real axis. Nevertheless, because of the pole at the origin, there needs to be a small detour of radius $\epsilon$ around the origin as shown.
The contour integral is then equal to
$$\int_0^{\pi} d\theta \, \frac{\cos{\theta}}{a+b \cos{\theta}} - i \, PV \int_{-1}^1 \frac{dx}{x} \, \frac{x^2+1}{b x^2+2 a x+b} - i (i \epsilon) \int_{\pi}^0 d\phi \, e^{i \phi} \, \frac1{\epsilon \, e^{i \phi}} \frac{\epsilon^2 e^{i 2 \phi}+1}{b \epsilon^2 e^{i 2 \phi}+ 2 a \epsilon \, e^{i \phi} + b} $$
The first integral is what we seek (for now). The third integral is, in the limit as $\epsilon \to 0$, $-\pi/b$. The second integral, the principal value integral, may be evaluated as follows:
$$\begin{align} PV \int_{-1}^1 \frac{dx}{x} \, \frac{x^2+1}{b x^2+2 a x+b} &= \frac1{b} \, PV \int_{-1}^1 \frac{dx}{x}\, \left (1 - \frac{2 a x}{b x^2+2 a x+b} \right ) \\ &= \frac1{b} \, PV \int_{-1}^1 \frac{dx}{x} - \frac{2 a}{b} \int_{-1}^1 \frac{dx}{b x^2+2 a x+b}\end{align}$$
Note that the first principal value integral on the RHS vanishes by symmetry. The second integral on the right needs not be expressed using principal value notation because the pole at the origin is removed. Accordingly,
$$\begin{align} PV \int_{-1}^1 \frac{dx}{x} \, \frac{x^2+1}{b x^2+2 a x+b} &= - \frac{2 a}{b} \int_{-1}^1 \frac{dx}{b x^2+2 a x+b} \\ &= -\frac{2 a}{b^2} \int_{-1}^1 \frac{dx}{\left ( x+\frac{a}{b} \right )^2 + 1-\frac{a^2}{b^2}}\\ &= -\frac{2 a}{b^2} \frac1{\sqrt{1-\frac{a^2}{b^2}}} \left [ \arctan{\left ( \frac{x+\frac{a}{b}}{\sqrt{1-\frac{a^2}{b^2}}} \right )} \right ]_{-1}^1 \\ &= \frac{\pi}{b} \frac{a}{\sqrt{b^2-a^2}} \end{align}$$
For convenience later on, we may write
$$ PV \int_{-1}^1 \frac{dx}{x} \, \frac{x^2+1}{b x^2+2 a x+b} = -i \frac{\pi}{b} \frac{a}{\sqrt{a^2-b^2}} $$
This way, we may write that the contour integral is equal to
$$\int_0^{\pi} d\theta \, \frac{\cos{\theta}}{a+b \cos{\theta}} - \frac{\pi}{b} \frac{a}{\sqrt{a^2-b^2}} - \frac{\pi}{b} $$
By the residue theorem, the contour integral is also equal to $i 2 \pi$ times the sum of the residues of the poles of the integrand of the contour integral inside the contour $C$. In this case, the only pole inside the contour is at $z_+ = -\frac{a}{b} + \sqrt{\frac{a^2}{b^2}-1}$. Computing the residue at this pole, the resulting equation for the integral we seek for now is
$$\int_0^{\pi} d\theta \, \frac{\cos{\theta}}{a+b \cos{\theta}} - \frac{\pi}{b} \frac{a}{\sqrt{a^2-b^2}} - \frac{\pi}{b} = -\frac{2 \pi}{b} \frac{a}{\sqrt{a^2-b^2}}$$
or
$$\int_0^{\pi} d\theta \, \frac{\cos{\theta}}{a+b \cos{\theta}} = -\frac{\pi}{b} \left ( \frac{a}{\sqrt{a^2-b^2}} - 1 \right ) $$
Now we may use this result to determine $I_1$. Again, subbing $a=1 \pm i t$ and $b=-i r$, we get that
$$\begin{align} I_1 &= -\frac1{2 \pi} \sqrt{\frac{2}{\pi t}} \frac{\pi}{i r} \left ( \frac{1-i t}{\sqrt{(1-i t)^2+r^2}} - \frac{1+i t}{\sqrt{(1+i t)^2+r^2}} \right ) \\ &= \sqrt{\frac{2}{\pi t}} \frac1{r} \operatorname{Im}{\left (\frac{1+i t}{\sqrt{(1+i t)^2+r^2}} \right )}\end{align}$$
And with that, we are technically finished. But as someone who likes explicit results, I will take this a bit further and express the result as follows:
I have verified this in Mathematica numerically.