If f (z) = u + iv is an analytic function, thena)u is harmonic functionb)v is harmonic functionc)Both u and v are harmonic functionsd)Both u and v are not harmonic functionsCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Explanation:

To understand why option C is the correct answer, let's first define what it means for a function to be analytic.

An analytic function is a complex-valued function that is differentiable at every point within its domain. In other words, the function has a derivative at each point within its domain.

Given that f(z) = u + iv is an analytic function, we can conclude the following:

Harmonic Functions:

A harmonic function is a real-valued function that satisfies Laplace's equation, which states that the sum of the second-order partial derivatives of the function is equal to zero. In other words, if u is a harmonic function, then it satisfies ∇²u = 0, where ∇² is the Laplacian operator.

Similarly, if v is a harmonic function, then it satisfies ∇²v = 0.

Derivatives of Analytic Functions:

Since f(z) is an analytic function, it means that both u and v have continuous first-order partial derivatives. This allows us to use the Cauchy-Riemann equations.

The Cauchy-Riemann equations relate the partial derivatives of u and v. They state that if f(z) = u + iv is an analytic function, then the following conditions must hold:

1. ∂u/∂x = ∂v/∂y
2. ∂u/∂y = -∂v/∂x

From these equations, we can see that the real part (u) and the imaginary part (v) of an analytic function are related to each other.

Conclusion:

Based on the above information, we can conclude that both u and v are harmonic functions. This is because if u and v are related to each other through the Cauchy-Riemann equations, and f(z) is an analytic function, then both u and v must satisfy Laplace's equation.

Therefore, the correct answer is option C: Both u and v are harmonic functions.

Answer

Concept:
If a function f satisfies Laplace's equation ∇²f = 0, then f is said to be a harmonic function.
Calculation:
If f (z) = u + iv is an analytic function, then

Now

Both u and v are satisfying Laplace’s equation (∇²f = 0).
∴ Both u and v are harmonic functions.

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