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m208E1review

m208E1review - and the theorem in section 21 8 Determine...

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Math 208 Exam I Review 1. Exam I will take place in class Tuesday, February 19. The relevant parts of Brown and Churchill are Sections 1-25, and 28-30. 2. Please be able to state the following: the definition of e , the definition of the derivative , the theorems in sections 20 and 21, the definition of an differentiable function, the definition of an analytic function, the definition of an entire function, the definition of a harmonic function, the definition of a harmonic conjugate . 3. The following problems are aimed to assist your review. Actual exam may be different from these problems. 4. Find | 1+2 i 3 - i | and express 1+2 i 3 - i in the form x + iy (x, y real). 5. Find: (a) (16) 1 / 4 ; (b) ( 1 - i 1+ i ) 1 / 3 . 6. Find and sketch the set onto which the transformation w = z 4 maps the annulus A = { z : 2 < | z | < 3 , 0 < Argz < π/ 3 } . 7. Show that f ( z ) = z 2 + 1 is differentiable everywhere and evaluate f 0 ( z ). Do this two in ways: (a) Using the definition of the derivative; (b) Using the Cauchy-Riemann Equations
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Unformatted text preview: and the theorem in section 21. 8. Determine where f ( z ) = z 2 + 2 z + ¯ z 2 is diﬀerentiable and evaluate f ( z ). 9. Is the function f ( z ) = | z | 2 diﬀerentiable at 0? Is f analytic at 0? Show your work and explain. 10. Show that u ( x,y ) = xe x cos x-ye x sin y is harmonic and ﬁnd a harmonic conjugate of u . 11. Suppose f ( z ) = u ( x,y )+ iv ( x,y ) is analytic in a domain D . Find a harmonic conjugate w for v in D so that g ( z ) = v ( x,y ) + iw ( x,y ) is analytic in D . 12. Determine where f ( z ) = zImz is diﬀerentiable and evaluate f ( z ). Do this two ways: (a) Using the deﬁnition of the derivative; (b) Using the Cauchy-Riemann Equations. 13. Describe the behavior of e z = e x e iy as (a) x → -∞ ; (b) y → ∞ . 14. Evaluate log(-1-i ) and Log (-1-i )....
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