Which of the following function f (z), of the complex variable z , is NOT analytic at all the points of the complex plane?a)f (z) = z2b)f (z ) = logzc)f (z) = ezd)f (z ) = sinzCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Explanation:
Analytic functions are those functions that satisfy the Cauchy-Riemann equations. These equations define a set of necessary conditions for a function to be analytic. If a function does not satisfy these conditions, then it is not analytic.

Let's check each option one by one:

a) f(z) = z^2
This function satisfies the Cauchy-Riemann equations and is therefore analytic at all points in the complex plane.

b) f(z) = logz
This function is not analytic at z = 0, since the logarithm is not defined at that point. Therefore, this function is not analytic at all points in the complex plane.

c) f(z) = e^z
This function satisfies the Cauchy-Riemann equations and is therefore analytic at all points in the complex plane.

d) f(z) = sinz
This function satisfies the Cauchy-Riemann equations and is therefore analytic at all points in the complex plane.

Therefore, the correct answer is option 'B' - f(z) = logz is not analytic at all points in the complex plane.

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