Help with sinh and cosh

bookie88 · #1 $Old$ October 27th, 2009, 11:58 AM

I need help deriving the expression for:
(a) sinhz/z^2 = 1/z + summation from n=0 to infinity z^2n+1/(2n+3)!
0<|z|<infinity

(b)z^3cosh(1/z) = z/2 + z^3 + summation from n=1 to infinity (1/(2n+2)!)*(1/z^2n-1) 0<|z|<infinity

Im lost if anyone can help please!!!

CaptainBlack · #2 $Old$ October 27th, 2009, 12:14 PM

Quote:

Originally Posted by bookie88 $View Post$

I need help deriving the expression for:
(a) sinhz/z^2 = 1/z + summation from n=0 to infinity z^2n+1/(2n+3)!
0<|z|<infinity

$\sum_{n=0}^{\infty}\frac{z^{2n+1}}{(2n+3)!}=\frac{1}{z^2}\sum_{n=0}^{\infty}\frac{z^{2n+3}}{(2n+3)!}=\frac{1}{z^2}(\sinh(z)-z)$

CB

durham2 · #3 $Old$ October 27th, 2009, 03:25 PM

I was pretty close to what you got. Any idea about the cosh one. that is the one i have no answer to

CaptainBlack · #4 $Old$ October 27th, 2009, 11:57 PM

Quote:

Originally Posted by durham2 $View Post$

I was pretty close to what you got. Any idea about the cosh one. that is the one i have no answer to

Same basic idea as the other, multiply the summation through by some power of z until the exponent of z in each term is +/- what is in the factorial. Then you will need some extra terms to make the sum the power series for sinh or cosh which when the sum is replaced by the hyperbolic function (which will be cosh in this case) you will need to subtract out the few terms you had to add to complete the series.

CB