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MAT389_Problem_Set_2_sols.pdf-MAT389 Fall 2013, Problem Set 2

mat195s_2008_exam.pdf

MAT389 Fall 2013, Problem Set 3 Functions 3.1 For each of the functions deﬁned below, describe the domain of deﬁnition that is under- stood: (i) f ( z )= 1 z 2 +1 , (ii) f ( z ) = Arg 1 z , (iii) f ( z )= z z +¯ z , (iv) f ( z )= 1 1 -| z | 2 . Limits 3.2 Let f :Ω ⊂ C → C , and let z 0 be an interior point of Ω. Prove that the limit lim z → z 0 f ( z ) is unique (if it exists). Hint: write out what it means for lim z → z 0 f ( z )= α and lim z → z 0 f ( z )= β . Using this information, what can you say about | α - β | ? 3.3 The following statement is false: lim z → z 0 f ( z )= ∞ = ⇒ lim ( x,y ) → ( x 0 ,y 0 ) u ( x, y ) and lim ( x,y ) → ( x 0 ,y 0 ) v ( x, y ) exist Can you provide a counterexample? Derivatives 3.4 Apply the deﬁnition of derivative to prove that f ( z ) = Re z is nowhere diﬀerentiable. 3.5 Where is f ( z )= | z | 2 diﬀerentiable? 3.6 Let f denote the function whose values are f (0) = 0 and f ( z )= (¯ z ) 2 z , for z 6 =0 Show that the Cauchy-Riemann equations are satisﬁed at the point z = 0 but that the derivative of f fails to exist there. Hint: to prove that the derivative does not exist, calculate the limit in the deﬁnition when approaching z = 0 horizontally (Δ y = 0), and along the line y = x (Δ x =Δ y ). 3.7 Suppose u ( x, y ) and v ( x, y ) have ﬁrst-order partial derivatives with respect to x and y at some point z 0 =( x 0 ,y 0 ) 6 = (0 , 0). (i) Use the change of coordinates x = r cos θ,y = r sin θ and the chain rule to show that u r = u x cos θ + u y sin θ, u θ = - u x r sin θ + u y r cos θ, v r = v x cos θ + v y sin θ, v θ = - v x r sin θ + v y r cos θ. ( * ) at the point z = z 0 . (ii) Using these identities and the Cauchy-Riemann equations in rectangular coordinates, deduce the polar form of the Cauchy-Riemann equations : u r = 1 r v θ , 1 r u θ = - v r (iii) Solve the equations ( * ) for u x , u y , v x and v y to show that u x = u r cos θ - u θ sin θ r , u y = u r sin θ + u θ cos θ r , v x = v r cos θ - v θ sin θ r , v y = v r sin θ + v θ cos θ r . ( ** ) (iv) Use these identities to deduce the Cauchy-Riemann equations in rectangular coor- dinates from their polar form. (v) Prove that the derivative of f ( re iθ )= u ( r, θ )+ iv ( r, θ ) at z = z 0 can be expressed in any of the following two forms: f 0 ( z 0 ) = (cos θ 0 - i sin θ 0 ) h u r ( r 0 ,θ 0 )+ iv r ( r 0 ,θ 0 ) i = - i z 0 h u θ ( r 0 ,θ 0 )+ iv θ ( r 0 ,θ 0 ) i

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