In a complex function
\(f(x, y) = u(x, y) + i v(x, y),\)
\(i\) is the imaginary unit, and \(x, y, u(x, y)\) and \(v(x, y)\) are real.
If \(f(x, y)\) is analytic then which of the following equations is/are TRUE?
\(f(x, y) = u(x, y) + i v(x, y),\)
\(i\) is the imaginary unit, and \(x, y, u(x, y)\) and \(v(x, y)\) are real.
If \(f(x, y)\) is analytic then which of the following equations is/are TRUE?
- \(\frac {∂^2u}{∂x^2}+\frac {∂^2u}{∂y^2}=0\)
- \(\frac {∂^2v}{∂x^2}+\frac {∂^2v}{∂y^2}=0\)
- \(\frac {∂^2u}{∂x^2}+\frac {∂^2v}{∂y^2}=0\)
- \((\frac {∂u}{∂x})(\frac {∂v}{∂x})+(\frac {∂u}{∂y})(\frac {∂v}{∂y})=0\)
The Correct Option is A, B, D
Solution and Explanation
Top Questions on Complex Functions
Let \( \Omega \) be a non-empty open connected subset of \( \mathbb{C} \) and \( f: \Omega \to \mathbb{C} \) be a non-constant function. Let the functions \( f^2: \Omega \to \mathbb{C} \) and \( f^3: \Omega \to \mathbb{C} \) be defined by \[ f^2(z) = (f(z))^2 \quad {and} \quad f^3(z) = (f(z))^3, \quad z \in \Omega. \]
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