Complex Mappings

[monomocoso]

Determine which curves in the z -plane correspond to straight lines in the w -plane under the mapping w=e^z

1. Obtain the equation of the curve (in the form y = f(x)) such that u =k where k is a constant maps to (k non-zero)

2. Obtain the equation of the curve (in the form y=f(x)) such that v = k where k is a constant maps to (k non-zero)

3. Obtain the equation of the curve such that v = mu (this covers v=0, and u=0 is not bad if you get this one)
Draw all three of these, but for 1 and 2, draw only k=1 and k=-1

 

[SlipEternal]

What are you having trouble with? What have you figured out so far? What do you think you should do to answer these questions?

 

[monomocoso]

I have cos(p) e^x cos y + sin(p) e^x sin y = k for a constant k and some angle p, but I don't know how to proceed

 

[SlipEternal]

I must admit that I don't know Complex Mappings at all, so I am not sure what end result you are looking for. My guess would be to factor out an \(\displaystyle e^x\) then use the sum of angles formula for cos. Remember that cos is even and sin is odd, so you will wind up with \(\displaystyle e^x\cos{(y-p)} = k\). From there, you can solve for y.

Does that help at all?

 

[monomocoso]

It does help, but I could use a little more in case anyone else has any ideas.