380_sample_exam_solutions[1]

# 380_sample_exam_solutions[1] - Math 380 Solutions to Sample...

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

Math 380 Solutions to Sample Exam Problems 1. Find each of the following: (a) (2 + i )(3 + 4 i ) = 2 + 11 i (b) Arg (1 ° p 3 i ) = ° °= 3 (c) Im( 2+ i 3 ° i ) = 1 2 (d) cos i = ( e 1 + e ° 1 ) = 2 (or cosh 1 ) (e) the polar form of (1 ° p 3 i ) = 2 e ° i°= 3 (f) i 1 = 3 = ° e i ( °= 2+2 ) ± 1 = 3 = e i ( °= 6+2 k°= 3) : The three distinct values are e i°= 6 = ² p 3 + i ³ = 2 ; e i (5 °= 6) = ² ° p 3 + i ³ = 2 ; and e i (9 °= 6) = ° i: 2. Find all values of i 2 ° i : i 2 ° i = i 2 i ° i = ( ° 1) e i ( °= 2+2 )( ° i ) = ° e °= 2+2 ; k = 0 ; ± 1 ; ± 2 ; ::: 3. Let f ( z ) = 2 xy + iy 2 . Find all points where f 0 ( z ) exisits and determine f 0 ( z ) at those points. Is f analytic at any point? Explain. Solution We need to check the Cauchy-Riemann conditions. We have u = 2 xy and v = y 2 : Since the partial derivatives of these are everywhere continuous, f 0 ( z ) exists precisely where the C-R equation hold, namely where u x = v y and u y = ° v x : That is, where 2 y = 2 y , and 2 x = 0 : Thus the C-R conditions are met if and only if Re z = 0 : So f 0 ( z ) exists only on the imaginary axis.

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