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# University of Toronto - Mississauga MAT334S - Winter 2016 Assignment 2 - due Friday, January 29th, at the beginning of your tutorial Problem 1....

a2-16.dvi

Problem 1. Compute lim z2−2i , if it exists. (Caution: if you make the direct substitution z→1+i z2−2z+2

z = 1 + i, you will get 0/0 ...)
Problem 2. Compute limz→0

University of Toronto – Mississauga MAT334S – Winter 2016 Assignment 2 - due Friday, January 29th, at the beginning of your tutorial Problem 1. Compute lim z 1+ i z 2 2 i z 2 2 z +2 , if it exists. (Caution: if you make the direct substitution z = 1 + i , you will get 0 / 0 ...) Problem 2. Compute lim z 0 ( z ¯ z + ¯ z z ) , if it exists. (Of course, this time it does not: use the exponential form z = rexp ( ) of z to verify this!) Problem 3. Here we shall compute derivatives the long way (for a simple function), and the short way (for a complicated function). Compute f ( z 0 ) for f ( z ) = z 2 by using the deFnition directly (i.e. by computing the limit lim Δ z 0 f ( z 0 z ) f ( z 0 ) Δ z ). Use the rules of di±erentiation and the chain rule to compute f ( z ) for f ( z ) = ze z 2 . Here, the exponent of z = x + iy is deFned by Euler’s formula: exp ( x + iy ) = exp ( x ) · exp ( iy ) = e x (cos y + i sin y ). Problem 4. Show that f ( z ) = ¯ z is nowhere di±erentiable (i.e. there is no point z 0 C such that f ( z 0 ) exists). Problem 5. Check that f ( z ) = e z + z + 1 is an entire function by using the Cauchy-Riemann equations and su²cient conditions for di±erentiability. Problem 6. Let f ( z ) be a di±erentiable function. Suppose that Re f ( z ) = x 2 - y 2 + y if we substitute z = x + iy into f ( z ). By using the Cauchy-Riemann equations, reconstruct Im f ( z ), up to an additive constant c C . Problem 7. Let f ( re ) = u ( r, θ ) + iv ( r, θ ) be given in polar coordinates by u ( r, θ ) = r ( r 2 - 1) cos θ and v ( r, θ ) = r ( r 2 + 1) sin θ . ³ind all points in C at which the Cauchy-Riemann equations are satisFed for f ( z ). (Since here we put z = re , use the polar form of Cauchy- Riemann equations) 1

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