Answer to Question #235405 in Complex Analysis for Fozia Sayda

Question #235405

1/z(ez-1) at its poles


1
Expert's answer
2021-09-12T23:51:39-0400
f(z)=1z(ez1)f(z)=\dfrac{1}{z(e^z-1)}

The singularities are z=0z = 0 and z=2nπi,n=±1,±2,....z = 2nπi, n = ±1, ±2, . . . .


The singularity at z=0z = 0 is a pole of order 2 since z=0z = 0 is a zero of order 2 of z(ez1)z(e^z-1).


1z(ez1)=1z(z+z2/2!+z3/3!+...)\dfrac{1}{z(e^z-1)}=\dfrac{1}{z(z+z^2/2!+z^3/3!+...)}

=1z2(1+z/2!+z2/3!+...)=1z2g(z),g(0)0=\dfrac{1}{z^2(1+z/2!+z^2/3!+...)}=\dfrac{1}{z^2}g(z), g(0)\not=0

The singularities z=2nπi,n=±1,±2,....z = 2nπi, n = ±1, ±2, . . . . are simple poles since they are simple zeros of 1z(ez1).\dfrac{1}{z(e^z-1)}.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment