380_sample_fnl_solns-corrected[1]

380_sample_fnl_solns-corrected[1] - Math 380 Solutions to...

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Unformatted text preview: Math 380 Solutions to Sample Final Exam Problems 1. (a) (2 + 3 i )(4 & 5 i ) = 23 + 2 i (b) Arg ( & p 3 + i ) = 5 6 & (c) 2+3 i 1 & i = & 1 2 + 5 2 i so the real part equals & 1 = 2 : (d) sin( & + i ) = & i sinh 1 (e) & p 3 + i = 2 e 5 &i= 6 (f) i 1 = 2 = exp(( &i= 2 + 2 k&i ) = 2) = exp( &i= 4) exp( k&i ) = exp( &i= 4) = & 1 p 2 + 1 p 2 i 2. (1 + i ) i = exp( i log(1 + i )) = exp( i (ln p 2 + &i= 4 + 2 k&i )) = e (1 = 2) i ln 2 e & &= 4 & 2 k& 3. The set H = f x + iy : x > g so f ( H ) = f w = e x e iy : x > and & 1 < y < 1g = f w : j w j > 1 g : If we express the elements of H in polar form, we see that H = re i : r > and & &= 2 < < &= 2 ; g ( H ) = f w = ln r + i : r > and & &= 2 < < &= 2 g : Thus g ( H ) = f u + iv : &1 < u < 1 and & &= 2 < v < &= 2 g : This is an open horizontal strip centered around the real axis having width equal to &: 4. The function f ( z ) = x 2 & y 2 & 2 xyi is di/erentiable in the complex sense precisely where the Cauchy- Riemann equations are satis&ed. Note u x = v y , 2 x = & 2 x , x = 0 and u y = & v x , & 2 y = 2 y , y = 0 . Thus f ( z ) exists only at z = 0 : We have f (0) = u x (0 ; 0) + iv x (0 ; 0) = 0 : Since f is not di/erentiable throughout any open neighborhood of any point, f is not analytic anywhere....
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380_sample_fnl_solns-corrected[1] - Math 380 Solutions to...

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