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) = ezc)f(z) = sin zd)f(z) = log zCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Analytic Functions in Complex Analysis

An analytic function is a complex function that is differentiable at every point in its domain.

If a function is analytic at all the points in the complex plane, then it is called an entire function.

If a function is not analytic at any point in its domain, then it is called a non-analytic function.

Out of the given options, the function f(z) = log z is not analytic at all the points in the complex plane.

Explanation

The function f(z) = log z is not analytic at z = 0 and any other point where z is negative or zero.

The reason for this is that the complex logarithm is a multivalued function. For any non-zero complex number z, there are infinitely many complex numbers w such that ez = w. So, we define the complex logarithm as follows:

log z = ln |z| + i arg(z)

where arg(z) is any angle whose tangent is the imaginary part divided by the real part of z.

However, when z is negative or zero, arg(z) is not well-defined, and so log z is not analytic at these points.

Hence, the function f(z) = log z is not analytic at all the points in the complex plane.

Answer

Concept:

The complex function f(z) = u (x, y) + iv (x, y) is to be analytic if it satisfy the following two conditions of Cauchy-Reiman theorem.

are continuous function of x and y.
f(z) = log z

Here, the function f(z) is analytic except z = 0. Since the function is not defined for these two values.
But in question it is asked at all points hence f(z) = log z is not analytic at all points.

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