Show that $L^{-1}\{(s^2+a^2)^{-\frac{3}{2}}\}=\frac{t}{a}J_1(ax)$

Question

Question:

Assume that $J_\alpha(x)$ is defined as below:
$$J_\alpha(x)=\sum_{n=0}^{\infty} \frac{(-1)^n}{n!\, \Gamma(n+\alpha+1)}\left(\frac{x}{2}\right)^{2n+\alpha}.$$

(a) Prove that $\frac{d}{dx}(x^{-\alpha} J_\alpha(x))=-x^{-\alpha}J_{\alpha+1}(x)$;

(b) If $L\{J_0(ax)\}=\frac{1}{\sqrt {s^2+a^2}}$, show that $L^{-1}\{(s^2+a^2)^{-\frac{3}{2}}\}=\frac{x}{a}J_1(ax)$ .

I proved part (a). But about part (b), what I have is:

$$L^{-1}\{(s^2+a^2)^{-\frac{3}{2}}\}=L^{-1}\{({(s^2+a^2)^{-\frac{1}{2}}})^{3}\}=J_0(x)*J_0(x)*J_0(x).$$

How should I reach $\frac{t}{a}J_1(ax)$ from $J_0(x)*J_0(x)*J_0(x)$ ?

Answer 1

Well, let's write:

$$\mathscr{L}_x\left[\frac{x}{\text{a}}\cdot\mathscr{J}_\alpha\left(\text{b}\cdot x\right)\right]_{\left(\text{s}\right)}=\frac{1}{\text{a}}\cdot\mathscr{L}_x\left[x\cdot\mathscr{J}_\alpha\left(\text{b}\cdot x\right)\right]_{\left(\text{s}\right)}\tag1$$

Using the 'frequency-domain derivative' property of the Laplace transform:

$$\frac{1}{\text{a}}\cdot\mathscr{L}_x\left[x\cdot\mathscr{J}_\alpha\left(\text{b}\cdot x\right)\right]_{\left(\text{s}\right)}=-\frac{1}{\text{a}}\cdot\frac{\partial}{\partial\text{s}}\left\{\mathscr{L}_x\left[\mathscr{J}_\alpha\left(\text{b}\cdot x\right)\right]_{\left(\text{s}\right)}\right\}\tag2$$

Using the 'time scaling' property of the Laplace transform:

\begin{equation} \begin{split} \text{Y}\left(\text{s}\right)&=-\frac{1}{\text{a}}\cdot\frac{\partial}{\partial\text{s}}\left\{\mathscr{L}_x\left[\mathscr{J}_\alpha\left(\text{b}\cdot x\right)\right]_{\left(\text{s}\right)}\right\}\\ \\ &=-\frac{1}{\text{a}}\cdot\frac{\partial}{\partial\text{s}}\left\{\frac{1}{\text{b}}\cdot\mathscr{L}_x\left[\mathscr{J}_\alpha\left(x\right)\right]_{\left(\frac{\text{s}}{\text{b}}\right)}\right\}\\ \\ &=-\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\frac{\partial}{\partial\text{s}}\left\{\mathscr{L}_x\left[\mathscr{J}_\alpha\left(x\right)\right]_{\left(\frac{\text{s}}{\text{b}}\right)}\right\} \end{split}\tag3 \end{equation}

Using the definition of the Bessel function:

\begin{equation} \begin{split} \text{Y}\left(\text{s}\right)&=-\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\frac{\partial}{\partial\text{s}}\left\{\mathscr{L}_x\left[\mathscr{J}_\alpha\left(x\right)\right]_{\left(\frac{\text{s}}{\text{b}}\right)}\right\}\\ \\ &=-\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\sum_{\text{n}=0}^\infty\frac{\left(-1\right)^\text{n}}{\text{n}!}\cdot\frac{1}{\Gamma\left(1+\alpha+\text{n}\right)}\cdot\frac{\partial}{\partial\text{s}}\left\{\mathscr{L}_x\left[\left(\frac{x}{2}\right)^{\alpha+2\text{n}}\right]_{\left(\frac{\text{s}}{\text{b}}\right)}\right\}\\ \\ &=-\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\sum_{\text{n}=0}^\infty\frac{\left(-1\right)^\text{n}}{\text{n}!}\cdot\frac{1}{\Gamma\left(1+\alpha+\text{n}\right)}\cdot\frac{\partial}{\partial\text{s}}\left\{\frac{1}{2^{\alpha+2\text{n}}}\cdot\frac{\Gamma\left(1+\alpha+2\text{n}\right)}{\left(\frac{\text{s}}{\text{b}}\right)^{1+\alpha+2\text{n}}}\right\}\\ \\ &=-\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\sum_{\text{n}=0}^\infty\frac{1}{2^{\alpha+2\text{n}}}\cdot\frac{\left(-1\right)^\text{n}}{\text{n}!}\cdot\frac{\Gamma\left(1+\alpha+2\text{n}\right)}{\Gamma\left(1+\alpha+\text{n}\right)}\cdot\frac{\partial}{\partial\text{s}}\left\{\frac{1}{\left(\frac{\text{s}}{\text{b}}\right)^{1+\alpha+2\text{n}}}\right\}\\ \\ &=-\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\sum_{\text{n}=0}^\infty\frac{1}{2^{\alpha+2\text{n}}}\cdot\frac{\left(-1\right)^\text{n}}{\text{n}!}\cdot\frac{\Gamma\left(1+\alpha+2\text{n}\right)}{\Gamma\left(1+\alpha+\text{n}\right)}\cdot\left(-\frac{1+\alpha+2\text{n}}{\text{s}}\cdot\frac{1}{\left(\frac{\text{s}}{\text{b}}\right)^{1+\alpha+2\text{n}}}\right)\\ \\ &=\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\sum_{\text{n}=0}^\infty\frac{1}{2^{\alpha+2\text{n}}}\cdot\frac{\left(-1\right)^\text{n}}{\text{n}!}\cdot\frac{\Gamma\left(2+\alpha+2\text{n}\right)}{\Gamma\left(1+\alpha+\text{n}\right)}\cdot\frac{1}{\text{s}}\cdot\frac{1}{\left(\frac{\text{s}}{\text{b}}\right)^{1+\alpha+2\text{n}}} \end{split}\tag4 \end{equation}

When $\Re\left(\alpha+2\text{n}\right)>-1\space\wedge\space\Re\left(\text{s}\right)>0$.

Now, when $\alpha=1$:

$$\mathscr{L}_x\left[\frac{x}{\text{a}}\cdot\mathscr{J}_1\left(\text{b}\cdot x\right)\right]_{\left(\text{s}\right)}=$$ $$\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\sum_{\text{n}=0}^\infty\frac{1}{2^{1+2\text{n}}}\cdot\frac{\left(-1\right)^\text{n}}{\text{n}!}\cdot\frac{\Gamma\left(2+1+2\text{n}\right)}{\Gamma\left(1+1+\text{n}\right)}\cdot\frac{1}{\text{s}}\cdot\frac{1}{\left(\frac{\text{s}}{\text{b}}\right)^{1+1+2\text{n}}}=$$ $$\frac{1}{\text{a}}\cdot\frac{1}{\text{b}}\cdot\sum_{\text{n}=0}^\infty\frac{1}{2^{1+2\text{n}}}\cdot\frac{\left(-1\right)^\text{n}}{\text{n}!}\cdot\frac{\Gamma\left(3+2\text{n}\right)}{\Gamma\left(2+\text{n}\right)}\cdot\frac{1}{\text{s}}\cdot\frac{1}{\left(\frac{\text{s}}{\text{b}}\right)^{2+2\text{n}}}\tag5$$

Now, when $\text{a}=\text{b}$:

$$\mathscr{L}_x\left[\frac{x}{\text{a}}\cdot\mathscr{J}_1\left(\text{a}\cdot x\right)\right]_{\left(\text{s}\right)}=\frac{1}{\text{s}^3}\cdot\frac{1}{\left(1+\left(\frac{\text{a}}{\text{s}}\right)^2\right)^\frac{3}{2}}\tag6$$

Now, I let you prove:

$$\frac{1}{\text{s}^3}\cdot\frac{1}{\left(1+\left(\frac{\text{a}}{\text{s}}\right)^2\right)^\frac{3}{2}}=\frac{1}{\left(\text{s}^2+\text{a}^2\right)^\frac{3}{2}}\tag7$$