Went to office hours today, it turns out for this question, it wasn't limited to just the residue theorems and Laurent series expansion methods. I tried using Cauchy integral formula, where z0 = 1/2 and f(z) = (z^10 / (z^10 + 2)) and got 2πi / 2049 as the final answer.
Homework Statement
Evaluate the integral using any method:
∫C (z10) / (z - (1/2))(z10 + 2), where C : |z| = 1
Homework Equations
∫C f(z) dz = 2πi*(Σki=1 Resp_i f(z)
The Attempt at a Solution
Rewrote the function as (1/(z-(1/2)))*(1/(1+(2/z^10))). Not sure if Laurent series expansion is the...
Okay, so if it's just the real part, iz+1 = i(x+iy) + 1 = ix - y +1 so the restriction would just be -y+1, where y ≠ 1?
I'm unsure what to do for a derivative, in my class notes it states that [log z ]' = 1/z so would it include the whole f(z) function, ie. ((z^2 + 2z + 5) / (iz+1))
Homework Statement
Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.)
f(z) = Log(iz+1) / (z^2+2z+5)
Homework EquationsThe Attempt at a Solution
Not sure how to even attempt this solutions but I wrote down that...
Homework Statement
Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1
Homework Equations
|z| = sqrt(x2 + y2)
z = x + iy
The Attempt at a Solution
Tried to put absolute values on every thing by the Triangle inequality
|z| - |1| + |i| = |1|...
You would get 1.003003 so its approximately 1. Curious question, why does this approximation matter to the proof or derivation of the original question?
I think \gamma = Cp/Cv and R = Cp - Cv.
I used an entropy equation and made it equal to 0:
0 = Cp*ln(T2/T1) - R*ln(p2/p1) and got (R/Cp)*ln(p2/p1) = ln(T2/T1). So I solved for R/Cv = \gamma -1 / \gamma... and the final result was
(P2/P1)^((\gamma -1) / \gamma) = T2/T1
Am I on the right...
Homework Statement
Show that \frac{dp}{p} =\frac{\gamma}{\gamma-1}\frac{dT}{T} if the decrease in pressure is due to an adiabatic expansion.Homework Equations
Poisson equations:
Pv^{\gamma}
Tv^{\gamma - 1}
Ideal Gas Law:
Pv=R_{d}T, where R_{d} is the dry air gas constant.
Hydrostatic...