Generating Function and Integral Representation of the Bessel Function

 
  In general, the Bessel function is expressed as infinite series, since we solve the Bessel equation using the Frobenious method ordinarily. However, we can write the Bessel function not only in the infinite series form, but also in an integral representation.

  In order to obtain the integral form of the Bessel function, we should figure out generating function of the Bessel function beforehand.

Generating Function of the 1st Kind Bessel Function

Proposition. The function below generates the 1st kind Bessel function as \[ \exp \left[\frac{x}{2} \left(h-\frac{1}{h}\right) \right] = \sum_{n=-\infty}^{\infty} { J_{n}(x) \, h^n } \; . \] proof. First, let us expand the exponential function as \[ \exp \left[ \frac{x}{2} \left(h-\frac{1}{h}\right) \right] = e^{\frac{xh}{2}} e^{-\frac{x}{2h}} = \sum_{r=0}^{\infty} { \frac{1}{r!} \left( \frac{xh}{2} \right)^r } \cdot \sum_{s=0}^{\infty} { \frac{1}{s!} \left( - \frac{x}{2h} \right)^s } = \sum_{r=0}^{\infty} \sum_{s=0}^{\infty} { (-1)^{s} \left( \frac{x}{2} \right)^{r+s} \frac{h^{r-s}}{r!\,s!} } \; . \]
Replace $n$ by $n=r-s$. Then, \[ \exp \left[ \frac{x}{2} \left(h-\frac{1}{h}\right) \right] = \sum_{n=-\infty}^{\infty} h^n \,\left[ \, \sum_{s=0}^{\infty} (-1)^s \frac{1}{s!\,(n+s)!} \left(\frac{x}{2} \right)^{2s+n} \, \right] \; . \]
Note that the $s$ summation term inside the square brackets above is identical to the usual infinite series representation of 1st kind Bessel function!

Therefore, \[ \therefore \qquad \exp \left[\frac{x}{2} \left(h-\frac{1}{h}\right) \right] = \sum_{n=-\infty}^{\infty} J_{n}(x) \, h^n \qquad \qquad \blacksquare \]

Integral Representation of the 1st Kind Bessel Function

Propostion. The 1st kind Bessel Function can be written as \[ J_{n}(x) = \frac{1}{n} \int_{0}^{\pi} d\theta \, \cos(x\sin{\theta} - n\theta) \quad \forall x \in \mathbb{R} \; , \; \forall n \in \mathbb{Z} \; . \] proof. Consider an integral of the generating function on complex plane along a closed contour \[ \oint_{C} \frac{e^{\frac{x}{2}(z-\frac{1}{z})}}{z^{n+1}} \; , \] where $C$ is a positively oriented unit circle centered at the origin.

Let $z=e^{i\theta}$ where $\theta$ runs from $0$ to $2\pi$. Then, $\; dz = ie^{i\theta}d\theta \; $ and $\; z-z^{-1} = 2i\sin{\theta} $. By changing the variable, we obtain \[ \oint_{C}\frac{e^{\frac{x}{2}(z-\frac{1}{z})}}{z^{n+1}} \;=\; \int_{0}^{2\pi} d\theta \, ie^{i\theta} \frac{e^{ix \sin{\theta}}}{e^{i(n+1)\theta}} \;=\; i \int_{0}^{2\pi} d\theta \, \exp{(ix \sin{\theta} - in\theta)} \; . \]
Note also that, \[ \oint_{C} \frac{e^{\frac{x}{2}(z-\frac{1}{z})}}{z^{n+1}} \;=\; \oint_{C} dz \sum_{m=-\infty}^{\infty} J_m(x)\, z^{m-n-1} \; . \]
Split the integral above into three parts as below. \[ \oint_{C} dz \sum_{m=-\infty}^{\infty} J_m(x)\, z^{m-n-1} \;=\; \oint_{C} dz \sum_{m=-\infty}^{n-1} \frac{J_{m}(x)}{z^{n-m+1}} \;+\; \oint_{C} dz \sum_{m=n+1}^{\infty} J_m(x)\, z^{m-n-1} \;+\; \oint_{C} dz\, \frac{J_{n}(x)}{z} \]
For the right hand side, observe that both integrands inside the first and the second integrals are holomorphic on and inside the integration contour $C$. Hence, both first and second term vanish! The third term is, however, not holomorphic anywhere and has a simple pole at $z=0$.

Applying the Cauchy residue theorem, we obtain \[ \oint_{C} \frac{e^{\frac{x}{2}(z-\frac{1}{z})}}{z^{n+1}} \;=\; 2\pi i \, J_{n}(x) \; . \]
Consequently, \[ \oint_{C} \frac{e^{\frac{x}{2}(z-\frac{1}{z})}}{z^{n+1}} \;=\; 2\pi i \, J_{n}(x) \;=\; i \int_{0}^{2\pi} d\theta \, \exp{(ix \sin{\theta} - in\theta)} \; . \]
Note that $ J_{n}(x) $ is a real-valued function. Hence, by picking only imaginary parts, we obtain \[ J_{n}(x) \;=\; \frac{1}{2\pi} \int_{0}^{2\pi} d\theta \, \cos{(x\sin{\theta} - n\theta)} \; . \]
From the even symmetry of $ \, \cos{(x\sin{\theta}-n\theta)} \, $ w.r.t. $ \, \theta=\pi \,$, we can conclude that \[ \therefore \qquad J_{n}(x) = \frac{1}{\pi} \int_{0}^{\pi} d\theta \, \cos{(x\sin{\theta}-n\theta) } \qquad \qquad \blacksquare \]

References

• George B. Arfken, Mathematical Methods for Physicists, Academic Press

• K. F. Riley, Mathematical Methods for Physics and Engineering, Cambridge University Press