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 $$ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad \text { and } \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} $$ (a) $u=x^{2}-y^{2}$ $$ v=2 x y $$ (b) $u=e^{x} \cos y$ $v=e^{x} \sin y$ (c) $u=\ln \left(x^{2}+y^{2}\right), \quad v=2 \tan ^{-1}(y / x)$

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

Key Concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of two partial differential equations that serve as a necessary (and under appropriate conditions, sufficient) condition for a function of a complex variable to be complex differentiable, or analytic. They establish a relationship between the partial derivatives of the real part u(x, y) and the imaginary part v(x, y) of the complex function f(z) = u(x, y) + iv(x, y), ensuring that u_x = v_y and u_y = -v_x.

Partial Derivatives

Partial derivatives involve differentiating a function with respect to one variable while keeping the other variables constant. In the context of verifying the Cauchy-Riemann equations, computing the partial derivatives of u and v with respect to x and y is essential to check whether the relations u_x = v_y and u_y = -v_x hold.

Analytic (Holomorphic) Functions

An analytic or holomorphic function is one that is complex differentiable at every point in an open region of the complex plane. A key property of such functions is that their real and imaginary parts satisfy the Cauchy-Riemann equations, which in turn implies that the functions can be locally expressed by a convergent power series.

Harmonic Functions

Harmonic functions are twice continuously differentiable functions that satisfy Laplace's equation. In complex analysis, if f(z) is analytic, its real and imaginary parts, u(x, y) and v(x, y), are harmonic. This property shows a deep connection between analytic functions and potential theory.

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