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**MAT334--Lectures & Home Assignments / Re: 2.5 Q29**

« **on:**November 14, 2018, 08:41:45 PM »

Hi I think this may help you a bit to understand it.

$\overline{G(\bar{z};u)} $

$=\overline{e^{(u/2)(\bar{z}-\frac{1}{\bar{z}})}}$

$=e^{(\frac{u}{2})(z-\frac{1}{\bar{z}})}$

$=G(z;u)$ if u is real.(to prove Hint)

Therefore if u is real,then $J_n(u)$ is real

Then given by (9) with $s=1:$

$J_n(u) = Re(J_n(u))$

$=Re(\int_{0}^{2\pi}e^{i(usin\theta - n\theta)}d\theta \frac{1}{2\pi})$

$=\int_{0}^{2\pi}cos(usin\theta - n\theta)d\theta\frac{1}{2\pi}$

$\overline{G(\bar{z};u)} $

$=\overline{e^{(u/2)(\bar{z}-\frac{1}{\bar{z}})}}$

$=e^{(\frac{u}{2})(z-\frac{1}{\bar{z}})}$

$=G(z;u)$ if u is real.(to prove Hint)

Therefore if u is real,then $J_n(u)$ is real

Then given by (9) with $s=1:$

$J_n(u) = Re(J_n(u))$

$=Re(\int_{0}^{2\pi}e^{i(usin\theta - n\theta)}d\theta \frac{1}{2\pi})$

$=\int_{0}^{2\pi}cos(usin\theta - n\theta)d\theta\frac{1}{2\pi}$