(a) It is known that a complex function f(2) = u(x, y) + iv(x, y) is differentiable at a point 20 e C if and only if the Cauchy-Riemann equations hold at zo and the first partial derivatives of u and v are continuous at zo. Use this to find all points of C at which f(2) = x - 3y? + 2x + i(3x?y – y + 2y). is differentiable, and find the derivatives at these points. (b) Determine now all points of C at which the function f in part (a) is analytic.

(a) It is known that a complex function f(2) = u(x, y) + iv(x, y) is differentiable at a point 20 e C if and only if the Cauchy-Riemann equations hold at zo and the first partial derivatives of u and v are continuous at zo. Use this to find all points of C at which f(2) = x - 3y? + 2x + i(3x?y – y + 2y). is differentiable, and find the derivatives at these points. (b) Determine now all points of C at which the function f in part (a) is analytic.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The complex function f(z)=u(x,y)+iv(x,y) can be differentiated at any point in the complex space. The real part of this function is given as u(x,y)=e−x[xsin(y)−ycos(y)].Determine the derivative of the function f(z) at z=i.
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Let w = f(z) = z2 + 3z=u+iv where, z is a complex variable such that z=x+ iy then, the value of v at z=1+3i is
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