Question

Zeige, daß die folgenden Funktionenpaare \((u, v)\) den CauchyRiemannschen Differentialgleichungen genügen: a) \(u(x, y)=x^{3}-3 x y^{2} ; v(x, y)=3 x^{2} y-y^{3}\); b) \(u(x, y)=\sin x \cdot \cosh y ; v(x, y)=\cos x \cdot \sinh y\).

Short answer

For both the cases: a) \(u(x, y)=x^{3}-3 x y^{2} ; v(x, y)=3 x^{2} y-y^{3}\) and b) \(u(x, y)=\sin x \cdot \cosh y ; v(x, y)=\cos x \cdot \sinh y\), the pairs of functions satisfy the Cauchy-Riemann equations.

Step-by-step solution

Solve for part a

We start by finding the derivatives of \(u(x, y)\) and \(v(x, y)\) for part a. Let us calculate the partial derivatives of the functions \(u(x, y)=x^{3}-3 x y^{2}\) and \(v(x, y)=3 x^{2} y-y^{3}\): \(\frac{\partial u}{\partial x}=3 x^{2}-3 y^{2}\) and \(\frac{\partial u}{\partial y}=-6 x y\). For \(v(x, y)\), we get \(\frac{\partial v}{\partial x}=6 x y\) and \(\frac{\partial v}{\partial y}=3 x^{2}-3 y^{2}\). Comparing \(\frac{\partial u}{\partial x}\) with \(\frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y}\) with -\(\frac{\partial v}{\partial x}\), we can see that they are equal, hence \(u(x, y)\) and \(v(x, y)\) satisfy the Cauchy-Riemann equations.

Solve for part b

Moving to part b, let us find the partial derivatives of the given functions \(u(x, y)=\sin x \cdot \cosh y\) and \(v(x, y)=\cos x \cdot \sinh y\). For \(u(x, y)\), we get \(\frac{\partial u}{\partial x}=\cos x \cdot \cosh y\) and \(\frac{\partial u}{\partial y}=\sin x \cdot \sinh y\). For \(v(x, y)\), we have \(\frac{\partial v}{\partial x}=-\sin x \cdot \sinh y\) and \(\frac{\partial v}{\partial y}=\cos x \cdot \cosh y\). Comparing the partial derivatives, we can see that again \(\frac{\partial u}{\partial x}\) equals \(\frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y}\) equals -\(\frac{\partial v}{\partial x}\). Hence, for part b, the given functions also satisfy the Cauchy-Riemann equations.

Key concepts

Complex Analysis

Complex Analysis is a fascinating branch of mathematics that focuses on functions of complex numbers. The key object of study in complex analysis is the holomorphic function, which is a function that's complex differentiable in a neighborhood around every point within its domain. Complex analysis skims over a beautiful harmony between algebra and geometry in two dimensions, providing powerful insights into various fields of mathematics and engineering.
The fundamental component in complex analysis is the complex plane, where every point represents a complex number. Complex analysis involves studying the properties of complex functions, which can often be elegantly solved using techniques that are otherwise difficult in real analysis.
- **Holomorphic Functions**: Functions that are complex differentiable. - **Cauchy-Riemann Equations**: A set of partial differential equations that provide conditions for a function to be holomorphic. Understanding these equations is critical in analyzing whether a complex function is holomorphic. - **Complex Integration**: Offers integral theorems like Cauchy's integral theorem, providing profound consequences in evaluating complex integrals.

Partial Derivatives

Partial derivatives play a crucial role in multi-variable calculus and are a key component in complex analysis. They involve taking the derivative of a multivariable function with respect to one variable, keeping the other variables constant. This technique is essential when dealing with functions of more than one variable, like the real and imaginary parts of a complex function.
In the context of the Cauchy-Riemann equations, partial derivatives depict how the functions' components change independently with respect to the complex plane's axes. The partial derivative operations required are as follows:
- **Respect to x**: Differentiating while keeping the other variable constant. This examines the rate of change across the real axis. - **Respect to y**: Keeps x constant, analyzing the rate of change along the imaginary axis.
In terms of the Cauchy-Riemann equations, if you have a function expressed as two parts, a real part, and an imaginary part, partial derivatives allow you to check if these parts satisfy the conditions of being a holomorphic function.

Holomorphic Functions

Holomorphic functions are central to the study of complex analysis. A function is termed holomorphic at a point if it's complex differentiable at that point and all nearby points. This continuity and differentiability lead to powerful properties, such as being infinitely differentiable and tightly conforming to their surrounding structure in the complex plane.
Holomorphic functions satisfy the Cauchy-Riemann equations, which form the bridge between real and complex representation of a function. These equations are - \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \] - \[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \] Where \(u\) and \(v\) are real-valued functions representing the real and imaginary parts of a complex function respectively.
When these equations are satisfied, the function can be said to be conformal, preserving angles and shapes to an infinitesimal degree around any given point. This remarkable property makes holomorphic functions useful tools in various applications, including electrical engineering and fluid dynamics.