Complex Integration Homework - Part (a) and (b) Help

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Homework Help Overview

The discussion revolves around a complex integration problem, specifically focusing on parts (a) and (b) of the assignment. The original poster mentions using the Cauchy integral formula for part (a) and expresses uncertainty about how it relates to part (b), which includes the term "hence." There is a suggestion that part (a) may provide insight into part (b).

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between parts (a) and (b), with one suggesting a substitution of variables to simplify the integral in part (b). There is an exploration of whether manipulations are necessary or if part (a) can directly assist in solving part (b).

Discussion Status

Some participants have offered guidance on potential substitutions and approaches, while others are still questioning the connections between the two parts of the problem. There is an acknowledgment of differing interpretations of how to proceed with part (b).

Contextual Notes

Participants are navigating the implications of the term "hence" in the problem statement, which suggests a direct link between the two parts. There is also a mention of avoiding unnecessary manipulations, indicating a desire to maintain clarity in the approach.

stripes
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Homework Statement



attachment.php?attachmentid=68696&stc=1&d=1397618882.jpg


Homework Equations





The Attempt at a Solution



I did part (a) which is pretty easy. Use the Cauchy integral formula. Part (b) says "hence", which leads me to believe that part (a) can be used for part (b), but I cannot see anything remotely related between the two. I don't want to do weird manipulations and substitutions if part (a) can help.

If someone could help me out with this I would really appreciate it.
 

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stripes said:

Homework Statement



attachment.php?attachmentid=68696&stc=1&d=1397618882.jpg


Homework Equations





The Attempt at a Solution



I did part (a) which is pretty easy. Use the Cauchy integral formula. Part (b) says "hence", which leads me to believe that part (a) can be used for part (b), but I cannot see anything remotely related between the two. I don't want to do weird manipulations and substitutions if part (a) can help.

If someone could help me out with this I would really appreciate it.

Just substitute ##e^{i \theta}## for ##z## in the integral and turn it into an integral ##d\theta##. Then use deMoivre.
 
Thanks Dick! Easy as pie now
 
Here is an elementary approach for part b) if you like:

Notice that the integral you seek is
$$\begin{aligned}
\Re\left(\int_0^{\pi} e^{k\cos t}e^{ik\sin t}\,dt\right) &= \Re\left(\int_0^{\pi}e^{ke^{it}}\,dt\right)\\
&=\Re\left(\int_0^{\pi} \left(1+ke^{it}+k^2e^{2it}+\cdots \right)\,dt\right)\\
&=\boxed{\pi}\\
\end{aligned}$$
 

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