Complex Variables Homework Solutions

[progrocklover]

Homework Statement

http://www.shotpix.com/images/83975343306312755574.jpg

Homework Equations

The Attempt at a Solution

For queation 1, I've found
w = u + jv = sinxcoshy + jcosxsinhy
u= sinxcoshy v=cosxsinhy
Then I put the 4 boundary line into sin(z),
I get
y=0 ==> w=sinx
x=pi/2 ==> w=coshy
x=-pi/2==>w=-coshy
x=0==>w=jsinhy
y=4==> u=sinxcosh4 v=cosxsinh4
Consider y=a, where a is a constant,
u^2 = (cosh^2 a) - v^2(cosh^2 a)/(sinh^2 a)
Hence, for y=4, u^2 = (cosh^2 4) - v^2(cosh^2 4)/(sinh^2 4)
but the problem is: I don't know how to sketch the curve of y=4


For question 5 and 6, I have no clue of how to get the Laurent expansions into the Region.
I just need some advice on how to do this type of question.(Any example of this type of question is appreicated)

For question 9iii), I actually got the answer, but the answer is rather long. I think there might be some better way to solve the problem.

I consider the integration of 1/2 z(e^jaz) / (z^4+b^4) from -finity to +infinity

In order to use 2(pi)j (Sum of Res z=zi f(z)), I again consider z^4 + b^4 = 0

and I've found the four roots(Let them be c,d,e,f, c and d are located on the upper half plane)

Then,

the integration of 1/2 z(e^jaz) / (z^4+b^4) from -finity to +infinity

= (pi)j (lim z-->c ze^jaz/(z-d)(z-e)(z-f) + lim z-->d ze^jaz/(z-c)(z-e)(z-f) )

Then i put the actual value of c,d,e,f into it, however, the final answer i got is really long.

 

[mighty2000]

My advice would be to try breaking down the problem into smaller parts and tackling them one at a time. For question 1, it looks like you have a good start by finding the boundary lines and their corresponding values for w. To sketch the curve of y=4, you can use the values you already found for u and v and plot them on a graph. Then connect the points to create the curve. For questions 5 and 6, it might help to review the definition and properties of Laurent series, and then try to apply them to the given functions. As for question 9iii), it sounds like you have the right idea by finding the roots and using the Residue Theorem, but it may be helpful to double check your calculations and see if there are any simplifications or shortcuts you can make. Additionally, it might be useful to look for similar examples or practice problems to get a better understanding of the concept. Overall, breaking down the problems and seeking out additional resources or examples can help you better understand and solve them.