Consider the complex valued function f(z) = 2z3 + |z|3 where z is a complex variable.The value of b for which the function f(z) is analytic is __________. (Answer up to the nearest integer)Correct answer is '0'. Can you explain this answer?

Question

Answer

Introduction:
In order to determine the value of b for which the function f(z) is analytic, we need to understand the concept of analyticity and apply it to the given function. Analyticity is a property of complex-valued functions that signifies the existence of a derivative at every point in their domain. According to the Cauchy-Riemann equations, a complex-valued function is analytic if its partial derivatives satisfy certain conditions.

Given Function:
The given function is f(z) = 2z^3 |z|^3, where z is a complex variable.

Derivation of Analyticity:
To determine the analyticity of the given function, we need to express it in terms of its real and imaginary parts. Let's assume z = x + iy, where x and y are real numbers.

Real and Imaginary Parts of f(z):
The real part of f(z) is given by Re[f(z)] = 2x^3 |z|^3, while the imaginary part is given by Im[f(z)] = 2y^3 |z|^3.

Partial Derivatives:
To check for analyticity, we need to calculate the partial derivatives of the real and imaginary parts of f(z) with respect to x and y.

Partial derivative with respect to x:
∂Re[f(z)]/∂x = 6x^2 |z|^3

Partial derivative with respect to y:
∂Re[f(z)]/∂y = 0

Partial derivative with respect to x:
∂Im[f(z)]/∂x = 0

Partial derivative with respect to y:
∂Im[f(z)]/∂y = 6y^2 |z|^3

Cauchy-Riemann Equations:
For a complex-valued function to be analytic, the partial derivatives must satisfy the Cauchy-Riemann equations:

∂Re[f(z)]/∂x = ∂Im[f(z)]/∂y
∂Re[f(z)]/∂y = -∂Im[f(z)]/∂x

Applying the Cauchy-Riemann Equations:
Comparing the partial derivatives calculated above, we can see that the Cauchy-Riemann equations are satisfied when y = 0. This means the function f(z) is analytic only when the imaginary part of z is zero, which implies the value of b is 0.

Conclusion:
The value of b for which the function f(z) = 2z^3 |z|^3 is analytic is 0.

Answer

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