Question

In: Advanced Math

What is the integral e^z/Z-1 dz where C is the circle|z| = 2?

What is the integral e^z/Z-1 dz where C is the circle|z| = 2?

Solutions

Expert Solution

Given curve |Z|=2 represents a circle with radius of 2 units also our complex integral has a singularity at z=1 ,so it must not be analytic at this point also this point lies inside the contour which is nice otherwise it's integral would have been 0. 

Let.h(z) be the integrand . Then h(z) = e^z /(z -1) . and z =1 liesinside it..More over e^z is analytic inside and on that circle. So by Cauchy’s integral formula

the integral over | z| = 2 is 2(pi) i x e^1 = 2(pi)e.


2iπe

Related Solutions

Complex Variable Evaluate the following integrals: a) int_c (z^2/((z-3i)^2)) dz; c=lzl=5 b) int_c (1/((z^3)(z-4))) dz ;...
Complex Variable Evaluate the following integrals: a) int_c (z^2/((z-3i)^2)) dz; c=lzl=5 b) int_c (1/((z^3)(z-4))) dz ; c= lzl =1 c) int_c (2(z^2)-z+1)/(((z-1)^2)(z+1)) dz ; c= lzl=3 (Details Please)
Use spherical coordinates to evaluate the triple integral ∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where...
Use spherical coordinates to evaluate the triple integral ∭e^−(x^2+y^2+z^2)/(x^2 + y^2 + z^2) dV , Where E is the region bounded by the spheres x^2 + y^2 + z^2 = 4 and x^2 + y^2 + z^2 = 9
Evaluate the line integral, where C is the given curve. ∫CF(x,y,z)⋅dr where F(x,y,z)=xi+yj+ysin(z+1)k and C consists...
Evaluate the line integral, where C is the given curve. ∫CF(x,y,z)⋅dr where F(x,y,z)=xi+yj+ysin(z+1)k and C consists of the line segment from (2,4,-1) to (1,-1,3).
remember and use <E>=?EiPiand Pi=1/Z ?exp(-?Ei) and Z=?exp(-?Ei) with ?=1/kT. (a) show that-dZ/d? = ?Eiexp(-?Ei) and...
remember and use <E>=?EiPiand Pi=1/Z ?exp(-?Ei) and Z=?exp(-?Ei) with ?=1/kT. (a) show that-dZ/d? = ?Eiexp(-?Ei) and (b) that the average energy can be derived directly from Z like <E>=-1/ZdZ/d? (c) show further that this simple expression is also correct <E> =-dlnZ/ d?
(1) z=ln(x^2+y^2), y=e^x. find ∂z/∂x and dz/dx. (2) f(x1, x2, x3) = x1^2*x2+3sqrt(x3), x1 = sqrt(x3),...
(1) z=ln(x^2+y^2), y=e^x. find ∂z/∂x and dz/dx. (2) f(x1, x2, x3) = x1^2*x2+3sqrt(x3), x1 = sqrt(x3), x2 = lnx3. find ∂f/∂x3, and df/dx3.
Hey. Given C = {z | z = z(t) = 10*e^(it), 0 <= t <= 2...
Hey. Given C = {z | z = z(t) = 10*e^(it), 0 <= t <= 2 pi}. How do I solve the following two integrals. Is there a way to do it with residues? a) f(z) = (cos(z) -1)/ z^3 b) f(z) = (sin(pi*z))/(z^3-1) Thank you!
consider the countour C given by |z+1|=1.5 oriented counterclockwise. Evaluate the integral_c (z^5+4z^3+9z^2+1+e^4z)/(z^k); where k= 1,2,3,4,5,6,7....
consider the countour C given by |z+1|=1.5 oriented counterclockwise. Evaluate the integral_c (z^5+4z^3+9z^2+1+e^4z)/(z^k); where k= 1,2,3,4,5,6,7. Their shoud be seven answers.
1.Evaluate the integral C where C is x=t^3 and y=t, 0 ≤ t ≤ 1 2.Find...
1.Evaluate the integral C where C is x=t^3 and y=t, 0 ≤ t ≤ 1 2.Find the area of the surface with vectorial equation r(u,v)=<u,u sinv, cu >, 0 ≤ u ≤ h, 0≤ v ≤ 2pi
1. Use integration by parts to evaluate the definite integral: ∫0.8 to 1.7 8ze^2z dz =...
1. Use integration by parts to evaluate the definite integral: ∫0.8 to 1.7 8ze^2z dz = 2. Evaluate ∫√cosx sin^3 x dx
1) Evaluate the integral from 0 to 1 (e^(2x) (x^2 + 4) dx) (a) What is...
1) Evaluate the integral from 0 to 1 (e^(2x) (x^2 + 4) dx) (a) What is the first step of your ‘new’ integral? (b) What is the final antiderivative step before evaluating? (c) What is the answer in simplified exact form? 2) indefinite integral (cos^2 2theta) / (cos^2 theta) dtheta (a) What is the first step of your ‘new’ integral? (b) What is the simplified integral before taking the antiderivative? (c) What is the answer in simplified form?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT