Math experts only. Fourier series for x+pi, and complex analysis, can you help?

Question

1) I need to find the Fourier series for f(x)=x+pi, -pi
2) For f(z)=((z-i)/(z+i))^(1/2)
a) Find the dominium where f(z) is analitic, is C-(o;-i)?, I mean the all coplez plane except the imaginary axis between o and -i.
b) The Taylor series at 2i, and it's convergence ratio. I was able to find it, but the derivatives was to complicated, I want to know if there is any trick for avoid those derivatives, such as to work on a simplier serie.
3) Only If you are a math genius: If the armonic function u(x,y) has a null in the x^2+y^2=1 unit circle, prove that is null in all its dominium,

Twopointers please abstain. Don´t make me lose my time, as you do. If you can't help, move on where you can.

Answer 1

Part 1

Do this in canonical form as the period is equal to 2pi. (I'm assuming that the period you meant is -pi
First thing is to split your integrals of ∫(x+pi)cos(mx) dx and ∫(x+pi)sin(mx) dx into seperate integrals of ∫x*cos(mx) dx + ∫pi*cos(mx) dx and ∫x*sin(mx) dx + ∫pi*sin(mx) dx.

You should get:

a0 = pi

as cos(0) = 1, ∫x term =0 but ∫pi term = pi

am = 0
bm = 2*((-1^(m+1))/m)

∫pi*cos(mx) dx and ∫pi*sin(mx) dx equal 0 because it is just a constant times cos or sin integrated over a integer number of periods. ∫x*cos(mx) dx and ∫x*sin(mx) dx are exactly as for the periodic identity function.

Are you sure you are only intergating over the period -pi
Part 2

Don't have an answer for that yet. If I get it, I'll put it up as an edit later.

P.S. Hope the integral sign displays ok, if not it may look a bit weird. Hope this helps.

Edit -

Sorry, can't help you with parts 2&3. The denominatorof f(z), √(z+i) clearly has a root at z=(0,-i) but other than that I'm not sure about the question. Hope the hints with part 1 were some help.

Answer 2

For part 1 you seem to have a very simple function that doesn't need a Fourier series. Are you sure that it is correctly written in the question?
For part 2a, the function is defined for C - (0,-i). This means the whole complex plane except the SINGLE point (0,-i) not the part of the imaginary plane between this and the origin. (You should any way have said that part of the imaginary axis from 0 to - 1, you didn't need the i because you have already said along that axis.)
I think that I can also give you some help with the rest of part 2.
You should find that the first derivative can be rearranged to
i*f(z)/(z + i)^2. (This may not be the case as I've done it rather quickly.) If so, this makes it easier to find the second derivative. The same thing might happen again but I haven't gone that far.

Answer 3

1q+1a=2p4>me.