Answer to Question #91402 in Calculus for Sajid

Question #91402
Q.Which of following function is analytic in comolex plane.
(i) f(z)=z ̅ (ii) f(z)=2x+ixy^2 (iii) f(z)= z^2
1
Expert's answer
2019-07-11T09:47:06-0400

A complex-valued function can be represented as "f(z)=u(x,y)+iv(x,y)" . "f(z)" is an analytic function in complex plane, if "u(x,y)" and "v(x,y)" satisfy Cauchy-Riemann equations at any point, i.e. "\\frac{\\partial u}{\\partial x}=\\frac{\\partial v}{\\partial y}" , "\\frac{\\partial u}{\\partial y}=-\\frac{\\partial v}{\\partial x}" .

(i) "f(z)=\\overline{z}=x-iy"

"\\frac{\\partial u}{\\partial x}=1", "\\frac{\\partial v}{\\partial y}=-1". The two partial derivatives are not equal. So "f(z)=\\overline{z}" is not analytic.

(ii) "f(z)=2x+ixy^2"

"\\frac{\\partial u}{\\partial x}=2", "\\frac{\\partial v}{\\partial y}=2xy". The two partial derivatives are not equal. So "f(z)=2x+ixy^2" is not analytic.

(iii) "f(z)=z^2=x^2-y^2+2ixy"

"\\frac{\\partial u}{\\partial x}=2x, \\frac{\\partial v}{\\partial y}=2x, \\frac{\\partial u}{\\partial y}=-2y, \\frac{\\partial v}{\\partial x}=2y."

So "u(x,y)" and "v(x,y)" satisfy Cauchy-Riemann equations at any point of complex plane, hence "f(z)=z^2" is analytic.

Answer: (iii) "f(z)=z^2"


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