University of Toronto  Mississauga MAT334S  Winter 2016 Assignment 2  due Friday, January 29th, at the beginning of your tutorial Problem 1....
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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 – MississaugaMAT334S – Winter 2016Assignment 2  due Friday, January 29th, at the beginning of your tutorialProblem 1. Compute limz→1+iz2−2iz2−2z+2, if it exists. (Caution: if you make the direct substitutionz= 1 +i, you will get 0/0 ...)Problem 2. Compute limz→0(z¯z+¯zz), if it exists. (Of course, this time it does not: use theexponential formz=rexp(iθ) ofzto verify this!)Problem 3. Here we shall compute derivatives the long way (for a simple function), and theshort way (for a complicated function).•Computef′(z0) forf(z) =z2by using the deFnition directly (i.e. by computing the limitlimΔz→0f(z0+Δz)−f(z0)Δz).•Use the rules of di±erentiation and the chain rule to computef′(z) forf(z) =zez2. Here,the exponent ofz=x+iyis deFned by Euler’s formula:exp(x+iy) =exp(x)·exp(iy) =ex(cosy+isiny).Problem 4. Show thatf(z) = ¯zis nowhere di±erentiable (i.e. there is no pointz0∈Csuchthatf′(z0) exists).Problem 5. Check thatf(z) =ez+z+ 1 is an entire function by using the CauchyRiemannequations and su²cient conditions for di±erentiability.Problem 6. Letf(z) be a di±erentiable function. Suppose that Ref(z) =x2y2+yif wesubstitutez=x+iyintof(z). By using the CauchyRiemann equations, reconstruct Imf(z),up to an additive constantc∈C.Problem 7. Letf(reiθ) =u(r, θ) +iv(r, θ) be given in polar coordinates byu(r, θ) =r(r21) cosθandv(r, θ) =r(r2+ 1) sinθ. ³ind all points inCat which the CauchyRiemannequations are satisFed forf(z). (Since here we putz=reiθ, use the polar form of CauchyRiemann equations)1
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