- #1

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Hi guys,

I'm pretty sure the following is true but I'm stuck proving it:

[tex]

\begin{align*}

\frac{1}{2\pi}\int_{-1}^1 \left(\frac{e^{\sqrt{1-y^2}}}{\sqrt{1-y^2}}+\frac{e^{-\sqrt{1-y^2}}}{\sqrt{1-y^2}}\right) e^{iyx} dy&=\frac{1}{2\pi i}\mathop\oint\limits_{|t|=1} \exp\left\{\sqrt{x^2-1}(t-1/t)/2\right\}t^{-1}dt\\

&=J_0(\sqrt{x^2-1})

\end{align*}

[/tex]

I tried the substitutions:

[tex]y=i/2(t-1/t)[/tex]

[tex]y=1/2(t-1/t)[/tex]

[tex]y=i/2\sqrt{x^2-1}(t-1/t)[/tex]

but not getting it. Also, I think only the first substitution converts the domain of integration into the required circle around the origin but that 's not too clear to me as well. Anyone can make a suggestion what to try next? Not sure if I should have posted this in the homework section.

Edit: Ok, I made a mistake in the Bessel integral notation. It's z/2 and not iz/2 so I changed it above and will re-do my calculations. May change things for me.

Thanks,

Jack

I'm pretty sure the following is true but I'm stuck proving it:

[tex]

\begin{align*}

\frac{1}{2\pi}\int_{-1}^1 \left(\frac{e^{\sqrt{1-y^2}}}{\sqrt{1-y^2}}+\frac{e^{-\sqrt{1-y^2}}}{\sqrt{1-y^2}}\right) e^{iyx} dy&=\frac{1}{2\pi i}\mathop\oint\limits_{|t|=1} \exp\left\{\sqrt{x^2-1}(t-1/t)/2\right\}t^{-1}dt\\

&=J_0(\sqrt{x^2-1})

\end{align*}

[/tex]

I tried the substitutions:

[tex]y=i/2(t-1/t)[/tex]

[tex]y=1/2(t-1/t)[/tex]

[tex]y=i/2\sqrt{x^2-1}(t-1/t)[/tex]

but not getting it. Also, I think only the first substitution converts the domain of integration into the required circle around the origin but that 's not too clear to me as well. Anyone can make a suggestion what to try next? Not sure if I should have posted this in the homework section.

Edit: Ok, I made a mistake in the Bessel integral notation. It's z/2 and not iz/2 so I changed it above and will re-do my calculations. May change things for me.

Thanks,

Jack

Last edited: