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

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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
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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...
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