1550251515-Classical_Complex_Analysis__Gonzalez_

602 Chapter^8

Fig. 8.10

(Fig. 8.10). Since

z z^2 zn

ez = 1 + - + - + · · · + - + · · ·

1! 2! n!

which converges for any finite value of z, we obtain

(

ez 1 1 1 1 1 n 3

f z) = z3 = z3 + z2 + 2! ; + ... + n! z - + ...

This is the required Laurent expansion for ez / z^3 , valid for z -:/: 0.

3. Apply formula (8.18-2) to show that the coefficients in the Laurent
   expansion in powers of z of the function
   f(z) = e(l/2)t(z-z-^1 )

valid for z -:/: 0 are given by

An(t) = -11'11" cos( nB - t sin B) d(}

7r 0

These coefficients, which are functions of the complex parameter t, are

known as the Bessel coefficients and are usually denoted Jn(t). Formula

(8.18-2) gives

__ 1_ J e<1/2)t(C-C

1

)

Jn(t) - 27ri cn+l d(

r

Choosing for r the circle ( = ei^9 (-7r ~ e ~ 7r), we get

(-C-^1 = ei^9 - e-i^9 = 2i sin(}

Hence

'Ir

J n ( ) t = ~ J e-in9+itsin^9 d(}

27r