CalcIVComplexSol

# CalcIVComplexSol - Solutions to sample questions for...

This preview shows pages 1–2. Sign up to view the full content.

Solutions to sample questions for complex integration 1. Let C be the curve z ( t ) = t 2 + it 3 for 0 t 1 , and let f ( z ) = e πiz . Compute C f ( z ) dz . We see right away that f ( z ) is holomorphic (since it’s the composition of functions we know to holomorphic), and thus we can use the fundamental theorem of calculus. The curve starts at 0 and ends at 1 + i , so we have C f ( z ) dz = - i π e πiz 1+ i 0 = i π ( 1 + e - π ) . 2. Let C be the line segment from i to 1, and let f ( z ) = z · z . Compute C f ( z ) dz . We have no reason to believe that f ( z ) is holomorphic (and, in fact, it isn’t as can be seen from the Cauchy-Riemann equations). So we compute the integral directly. We can parametrize C by z ( t ) = t + (1 - t ) i for 0 t 1. Then z ( t ) = 1 - i and f ( z ( t )) = 2 t 2 - 2 t + 1, after a little algebra. Thus C f ( z ) dz = 1 0 ( 2 t 2 - 2 t + 1 ) (1 - i ) dt = (1 - i ) 2 3 t 3 - t 2 + t 1 0 = 2 3 - 2 3 i 3. Let C be the curve consisting of the line segment from 1 2 - i to 2 - i , followed by the segment from 2 - i to 2 + i , followed by the segment from 2 + i to 1 2 + i , followed by the segment from 1 2 + i to 1 2 - i . If

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern