Complex Analysis: Theory and Applications · Problem 3.2.1

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · cauchy-riemann-differentiability

Problem

Use the Cauchy Riemann conditions to determine where the following functions are differentiable, and evaluate the derivatives at those points where they exist. (a) f(z)=iz+4if(z) = iz + 4i. (b) f(z)=f(x,y)=y+ixx2+y2f(z) = f(x, y) = \dfrac{y+ix}{x^2+y^2}. (c) f(z)=−2(xy+x)+i(x2−2y−y2)f(z) = -2(xy + x) + i(x^2 - 2y - y^2). (d) f(z)=x3−3x2−3xy2+3y2+i(3x2y−6xy−y3)f(z) = x^3 - 3x^2 - 3xy^2 + 3y^2 + i(3x^2y - 6xy - y^3). (e) f(z)=x3+i(1−y)3f(z) = x^3 + i(1-y)^3. (f) f(z)=z2+zf(z) = z^2 + z. (g) f(z)=x2+y2+i2xyf(z) = x^2 + y^2 + i2xy. (h) f(z)=∣z−(2+i)∣2f(z) = |z - (2+i)|^2.

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