In each part, show that u(x, y) and v(x, y) satisfy the...

In each part, show that $u(x, y)$ and $v(x, y)$ satisfy the Cauchy-Riemann equations $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$ $$\begin{array}{ll}{\text { (a) } u=x^{2}-y^{2},} & {v=2 x y} \\ {\text { (b) } u=e^{x} \cos y,} & {v=e^{x} \sin y} \\ {\text { (c) } u=\ln \left(x^{2}+y^{2}\right),} & {v=2 \tan ^{-1}(y / x)}\end{array}$$

In each part, show that $u(x, y)$ and $v(x, y)$ satisfy the Cauchy-Riemann equations
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$
$$\begin{array}{ll}{\text { (a) } u=x^{2}-y^{2},} & {v=2 x y} \\ {\text { (b) } u=e^{x} \cos y,} & {v=e^{x} \sin y} \\ {\text { (c) } u=\ln \left(x^{2}+y^{2}\right),} & {v=2 \tan ^{-1}(y / x)}\end{array}$$

Key Concepts

Cauchy-Riemann Equations

These are a set of two partial differential equations that relate the partial derivatives of the real and imaginary parts of a complex function. They serve as a necessary (and under certain conditions, sufficient) criterion for a function to be analytic (holomorphic). When a complex function is expressed as f(z) = u(x, y) + iv(x, y), the equations state that ?u/?x must equal ?v/?y and ?u/?y must equal -?v/?x, ensuring the complex derivative exists and is independent of the direction of approach.

Partial Differentiation

In the context of the Cauchy-Riemann equations, partial differentiation is used to compute the derivative of multivariable functions with respect to one variable while treating the other variable as constant. This process is essential for verifying the conditions for complex differentiability, as it allows us to isolate and compare the rates of change of the real and imaginary parts along the x and y directions.

Analytic Functions

An analytic or holomorphic function is a complex function that is differentiable at every point in an open subset of the complex plane. The satisfaction of the Cauchy-Riemann equations at every point in a domain is a key characteristic of such functions. Analytic functions enjoy numerous strong properties, including the existence of power series representations and the satisfaction of various integral theorems in complex analysis.

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