Q2Advanced Mathematics

Question

Show that the polar form of C-R equation's is. $\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial θ}, \frac{\partial v}{\partial r} = - \frac{1}{r} \frac{\partial u}{\partial θ}$ Using this result show that log z is analytic.

Answer 1

Step 1: Understand the Problem Statement Identify the core mathematical topic (e.g., Complex Analysis, Partial Differential Equations, Numerical Methods, or Special Functions) required to solve the given expression or theorem.

Step 2: Apply the Relevant Formula/Theorem Depending on the question, apply the standard RTU formula. For instance, if it's a Cauchy-Riemann equation problem, ensure $u_{x} = v_{y}$ and $u_{y} = - v_{x}$ . If it's a series solution, use Frobenius method. Write down the generalized equation: $f (z) = u (x, y) + i v (x, y)$ or the relevant integral form.

Step 3: Step-by-Step Derivation 1. Substitute the given boundary conditions or initial values into the generalized formula. 2. Perform integration, differentiation, or algebraic simplification as required. 3. Keep track of constants of integration (e.g., $C_{1}, C_{2}$ ) if solving a differential equation.

Step 4: Final Evaluation Simplify the expression to arrive at the final closed-form solution or numerical approximation. Ensure the final result is boxed or clearly stated as per examination standards.

Q2Advanced Mathematics

Question

Answer