253-hwk-33a

# 253-hwk-33a - j cos z j j sin z j> 1 in the complex plane...

This preview shows page 1. Sign up to view the full content.

HPHY 253 Homework 33a Complex Numbers Dr. A. E. Bak Name 1. Express the following complex numbers in both rectangular and polar form. ( a ) 1 1 + i ; ( b ) 3 + i 2 + i ; ( c ) i 4 ; ( d ) (1 + 2 i ) 3 ; ( e ) 1 + i p 3 p 2 + i p 2 ! 50 : Limit the phase range to 0 ° ° < 2 ± . 2. Solve for all possible values of the real numbers x and y in the following equations. ( a ) x + iy = 3 i ± 4 ; ( b ) ( x + iy ) 3 = ± 1 ; ( c ) x + iy x ± iy = ± i : 3. Arfken & Weber Problem 6 : 1 : 5 . Show that complex numbers have square roots and that the square roots are themselves complex numbers. What are the square roots of + i ? 4. Find all the values of the following roots. ( a ) 3 p 1 ; ( b ) 3 p 27 ; ( c ) 3 p ± 8 i ; ( d ) 4 p ± 1 ; ( e ) 5 p ± 1 ± i : Express your values in rectangular form. 5. Arfken & Weber Problem 6 : 1 : 10 . Using the identities cos z = e + iz + e ° iz 2 ; sin z = e + iz ± e ° iz 2 i ; which are established from comparison of the relevant power series, show that cos z = cos x cosh y ± i sin x sinh y ; sin z = sin x cosh y + i cos x sinh y ; and j cos z j 2 = cos 2 x + sinh 2 y ; j sin z j 2 = sin 2 x + sinh 2 y ; where z = x + iy in rectangular form. These relations demonstrate that we may have
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: j cos z j ; j sin z j > 1 in the complex plane. 6. Express each of the following complex numbers in rectangular form. ( a ) sin & 1 2 ; ( b ) cos & 1 & i p 8 ± ; ( c ) cosh & 1 ( ± 1) ; ( d ) sinh & 1 & i= p 2 ± : Note that cos ( iu ) = cosh u ; sin ( iu ) = + i sinh u for any real number u . 7. Arfken & Weber Problem 6 : 1 : 15 . Find all the zeros of the functions ( a ) sin z ; ( b ) cos z ; ( c ) sinh z; ( c ) cosh z : Note that z = x + iy in rectangular form. 8. Arfken & Weber Problem 6 : 1 : 17 . In the quantum theory of photoionization, we encounter the identity ² ia ± 1 ia + 1 ³ ib = exp ´ ± 2 b cot & 1 a µ ; where a and b are real numbers. Verify this identity. 1...
View Full 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