Hyperbolic function equations

• Apr 30th 2011, 03:55 AM
Glitch
Hyperbolic function equations
The question:
Find all solutions $z \in C$ of sinh(z) = 0

My attempt:
sinh(z) = sinh(x)cos(y) + icosh(x)sin(y)

Equating real and imaginary parts:
sinh(x)cos(y) = 0 (1)
cosh(x)sin(y) = 0 (2)

For (1):
sinh(x) = 0 when x = 0, and cos(y) = 0 when $y = (\frac{1}{2} + k)\pi; k \in Z$

Sub these values into (2):
cosh(0) = 1 (therefore, we need sin(y) = 0)
$sin((\frac{1}{2} + k)\pi)$is never 0.

So it appears that it never equals zero. However, my text states that it does when $z = ik\pi$ Where have I gone wrong? Thanks.
• Apr 30th 2011, 05:18 AM
chisigma
Setting z= i x is sinh z = i sin x so that the function vanishes for x= k $\pi$ -> z= i k $\pi$...

Kind regards

$\chi$ $\sigma$