A First Course in Complex Analysis · Problem 18

Examples of Functions — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchy-riemann-equations

Problem

Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable. (a) f(z)=e−xe−iyf(z) = e^{-x}e^{-iy} Hint. Use the Cauchy--Riemann equations (2.3). (b) f(z)=2x+ixy2f(z) = 2x + ixy^2 (c) f(z)=x2+iy2f(z) = x^2 + iy^2 (d) f(z)=exe−iyf(z) = e^x e^{-iy} (e) f(z)=cos⁡xcosh⁡y−isin⁡xsinh⁡yf(z) = \cos x \cosh y - i \sin x \sinh y (f) f(z)=Im⁡zf(z) = \operatorname{Im} z (g) f(z)=∣z∣2=x2+y2f(z) = |z|^2 = x^2 + y^2 (h) f(z)=zIm⁡zf(z) = z\operatorname{Im} z (i) f(z)=ix+1yf(z) = \dfrac{ix+1}{y} (j) f(z)=4(Re⁡z)(Im⁡z)−i(zˉ)2f(z) = 4(\operatorname{Re} z)(\operatorname{Im} z) - i(\bar{z})^2 (k) f(z)=2xy−i(x+y)2f(z) = 2xy - i(x+y)^2 (l) f(z)=z2−zˉ2f(z) = z^2 - \bar{z}^2

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