Question

The complex function f(z) = 1/(z^4 - 1) has poles at +-1 and +-i, which may or may not contribute to the closed curve integral around C of f(z)dz. In turn, the closed curve C that you use depends on the 2nd letter of your first name! Specifically, convert that letter to its numerical position in the Roman alphabet (A=1, B=2, ..., Z=26), then divide by 4. Don't worry about fractions, just save the REMAINDER which will be an integer 0, 1, 2, or 3 (this is called Modular Arithmetic and is a very big area of math, but not in 422). If your remainder is 0 then your contour is C0, which is the unit circle centered at z=1 in the complex plane. If your remainder is 1 your contour is C1 the unit circle around i, C2 = unit circle around -1, and C3 = unit circle around -i. All contours are in the widdershins direction, as usual. I would like you to calculate integral_C f(z)dz around your personal circle.

The 2nd letter: h

Answer #1

suppose that the only singularities of the function g(z) are
finitely many poles which lie away from the origin and the negative
real axis. show that integration of the function f(z)=g(z)lnz with
-pi<arg(z)<=pi , around an appropriate keyhole contour, leads
to being able to find the value of the integral g(-x)dx in terms of
the residues of f(z).

Complex Variable
Evaluate the following:
A) ∫_∣z−i∣=2 (2z+6)/(z^2+4) dz
B) ∫_∣z∣=2
1/((z−1)^2(z−3)) dz ( details
please)
C) ∫_∣z∣=1 e^(4/z) dz

Consider a function F=u+iv which is analytic on the set
D={z|Rez>1} and that u_x+v_y=0 on D. Show that there exists a
real constant p and a complex constant q such that F(z)=-ipz+q on
D.
Notation: Here u_x denotes the partial derivative of u with
respect to x and v_y denotes the partial derivative of v with
respect to y.

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