Intereting Posts

Can one construct a “Cayley diagram” that lacks only an inverse?
Inverse of the sum of a symmetric and diagonal matrices
A problem on Number theory
Solving $13\alpha \equiv 1 \pmod{210}$
How many triangles can be created from a grid of certain dimensions?
Why are projective morphisms closed?
Product of ideals corresponding to vanishing of points is equal to their intersection
A question on semifinite measures
How do we know that we found all solutions of a differential equation?
The general argument to prove a set is closed/open
If $\,\lim_{x\to 0} \Big(f\big({a\over x}+b\big) – {a\over x}\,f'\big({a\over x}+b\big)\Big)=c,\,$ find $\,\lim_{x\to\infty} f(x)$
What REALLY is the modern definition of Euclidean Spaces?
What is the intuition behind the Wirtinger derivatives?
How to construct a bijection from $(0, 1)$ to $$?
Expectation of Ito integral

How could it be proved that

$$ \int_0^\infty J_0\left(\alpha\sqrt{x^2+z^2}\right)\ \cos{\beta x}\ \mathrm{d}x = \frac{\cos\left(z\sqrt{\alpha^2-\beta^2}\right)}{\sqrt{\alpha^2-\beta^2}} $$

for $0 < \beta < \alpha$ and $z > 0$ ?

$J_0(x)$ is the zeroth order of Bessel function of the first kind.

I found this integral in Gradshteyn and Ryzhik’s book 7th edition, section 6.677, the equation number 3. Any helps and hints will be appreciated!

- $ y' = x^2 + y^2 $ asymptote
- What is the closed form of $\sum _{n=1}^{\infty }{\frac { {{\it J}_{0}\left(n\right)} ^2}{{n}^4}}$?
- Fourier Cosine Transform (Parseval Identity) for definite integral
- Difficult infinite integral involving a Gaussian, Bessel function and complex singularities
- The Relative Values of Two Modified Bessel Function of the First Kind
- Integral of Bessel function multiplied with sine

- Upper bound on integral: $\int_1^\infty \frac{dx}{\sqrt{x^3-1}} < 4$
- Proving $\int_{0}^{\pi/2}x\sqrt{\tan{x}}\log{\sin{x}}\,\mathrm dx=-\frac{\pi\sqrt{2}}{48}(\pi^2+12\pi \log{2}+24\log^2{2}) $
- Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$
- Closed-form of $\int_0^1 \frac{\ln^2(x)}{\sqrt{x(a-bx)}}\,dx$
- How can we prove that $2e^2\int_{0}^{\infty} {x\ln x\over 1-e^{2e\pi{x}}}\mathrm dx=\ln A?$
- Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$
- Evaluation of complete elliptic integrals $K(k) $ for $k=\tan(\pi/8),\sin(\pi/12)$
- Integrate $\cos^2(\pi x)\cos^2(\frac{n\pi}{x})$
- Arc length of logarithm function
- Tough Inverse Fourier Transform

Note that for $r>0$ one has integral representation

$$J_0(r)=\frac{1}{2\pi}\int_0^{2\pi}e^{ir\cos\phi}d\phi$$

Hence

$$I=\int_0^{\infty}J_0\left(\alpha\sqrt{x^2+z^2}\right)\cos \beta x\,dx=

\frac{1}{4\pi}\int_0^{2\pi}\int_{-\infty}^{\infty}e^{i\alpha\sqrt{x^2+z^2}\cos\phi}\cos\beta x \, d\phi.\tag{1}$$

On the other hand,

$$\sqrt{x^2+z^2}\cos\phi=z\cos(\phi-\phi_0)+x\sin(\phi-\phi_0),$$

where $\tan\phi_0=-\frac{x}{z}$. Exchanging the order of integration in (1) and shifting $\phi$ by $\phi_0$, we arrive at

$$I=\frac{1}{4\pi}\int_0^{2\pi}\int_{-\infty}^{\infty}e^{i\alpha(z\cos\phi+x\sin\phi)}\cos\beta x \, d\phi.$$

Finally, using that $\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\omega x}dx=\delta(\omega)$ we obtain

$$I=\frac{1}{4}\int_0^{2\pi}e^{i\alpha z\cos\phi}\Bigl[\delta\left(\alpha\sin\phi+\beta\right)+\delta\left(\alpha\sin\phi-\beta\right)\Bigr]d\phi$$

It remains to use $\delta(f(x))=\sum\limits_{\text{zeros of }f}\frac{1}{|f'(x_k)|}\delta(x-x_k)$ and compute the two contributions coming from each of the two delta-functions.

- Why is it that if I count years from 2011 to 2014 as intervals I get 3 years, but if I count each year separately I get 4 years?
- Finding the asymptotic behavior of the recurrence $T(n)=4T(\frac{n}{2})+n^2$ by using substitution method
- What is the algorithm hiding beneath the complexity in this paper?
- Proving $\phi(m)|\phi(n)$ whenever $m|n$
- Proof that if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \mid \phi(x) \geq \alpha \}$ is closed.
- Finding points on ellipse
- Show that ${(F_n^2+F_{n+1}^2+F_{n+2}^2)^2\over F_{n}^4+F_{n+1}^4+F_{n+2}^4}=2$
- Expanded concept of elementary function?
- Derivation of the general forms of partial fractions
- Derivation of the formula for Ordinary Least Squares Linear Regression
- How far is being star compact from being countably compact？
- Is Plancherel's theorem true for tempered distribution?
- Solving trigonometric equations of the form $a\sin x + b\cos x = c$
- $n^5-n$ is divisible by $10$?
- $a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt{abc} \leq 28$