- #1
degs2k4
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Homework Statement
The Attempt at a Solution
PART1)
Series of Taylor for e^(3z) + 2 (based on series for e^z)
[tex]
\sum_{n=0}^{\infty} \frac{(3z)^{n}}{n!}= (1+2) + 3z + \frac{(3z)^2}{2!} + \frac{(3z)^3}{3!} + ...
[/tex]
Series of Taylor for 3e^(z) (based on series for e^z)
[tex]
\sum_{n=0}^{\infty} \frac{3z^{n}}{n!}= 3 + 3z + \frac{z^2}{2!} + \frac{3z^3}{3!} + ...
[/tex]
Subtracting each term of the above 2 series, and dividing by z^3 we get the Laurent series:
[tex]
0 + 0 + \frac{6}{2!z} + \frac{24}{3!} + ...
[/tex]
Since the laurent series (negative powers) stops at n=1, the pole is of order 1.
the residue af f(z) at z=0 is 0, since z=0 is outside the contour C.
Questions:
1) Is this ok? if not, why?
2) Is there any other way of knowing a pole order in this function without using Laurent Series ? (eg by limits of derivates?) If not, we must know as much (Taylor)Laurent series as we can in order to solve those problems ?
3) If the both the numerator/denominator of the function were polynomial and z = a a pole of it, we only need to check the power of (z-a) in order to know the order of the pole?PART2)
The integral of the contour C is 0, as no singularities can be found inside C.
[tex]
\oint_C = 0 = \int_A + \int_B + \int_C + \int_D
[/tex]
I want to demonstrate that \int_D tends to 0 as R tends to infinite. For doing so I tried the following parametrization and check
the limit when R tends to 0, but couldn't reach something showing that the limit is 0.
[tex]
z = Re^{it}, t=0...\pi
[/tex]
[tex]
dz = iRe^{it} dt
[/tex]
I also want to demonstrate that \int_B tends to 0 as R tends to infinite, but can't see how to do it neither. Visually I can see
that when R tends to 0 the integrals over A and C become so big that the integral over B is negligible (zero).
Questions:
How could I demonstrate that those two integrals tend to to 0 (which I think they do) ?
Thanks in advance...