374chw7 - i i and they are all real. ( Hint : Remember the...

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HW#7 —Phys374—Spring 2008 Prof. Ted Jacobson Due before class, Friday, April 4, 2006 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374c/ jacobson@physics.umd.edu Fun with complex numbers 1. Express the following in “Cartesian form” x + iy , where x and y are real: 1 / (2 - 3 i ), (1 + 2 i ) / (3 + 4 i ), 5 e 6 i . [2+2+2=6 pts.] 2. Express the following in “polar form” re , where r is a real positive number and θ is real: - 6, - 5 i , (1 + i ) / 2, 2 - 3 i , (2 + i ) / (1 + 2 i ). ( Note : Be careful to get the correct sign for the phase.) [2+2+2+2+2=10 pts.] 3. (i) Find all the cube roots of - 1, i.e. ( - 1) 1 / 3 , and express them all in both polar form and in Cartesian form. (ii) Plot and label them in the complex plane. [10+5=15 pts.] 4. Show that there are infinitely many values of
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Unformatted text preview: i i and they are all real. ( Hint : Remember the denition of the complex exponential: w z = exp( z ln w ).) [5 pts.] 5. Prove the trigonometric identities cos( a + b ) = cos a cos b-sin a sin b and sin( a + b ) = sin a cos b +cos a sin b by taking the real and imaginary parts of the identity exp( i ( a + b )) = exp( ia ) exp( ib ). You may of course use the fact that exp( i ) = cos + i sin . [ 5pts.] 6. Express the real and imaginary parts of the following functions in terms of x = Re ( z ) and y = Im ( z ): z 3 , e z , e iz , sin z , 1 / ( z 2 + 1). [2+2+2+2+2=10 pts.] 7. Problem 16.1h [10 pts.]...
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