91Ó°ÊÓ

Let \(f\) be a complex-valued function defined on a disk \(D: \quad|z|0)\) which is differentiable there. Let $$ u(x, y)=(\operatorname{Re} f)(x+i y), \quad v(x, y)=(\operatorname{Im} f)(x+i y) $$ Show that $$ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} $$ for all \(z-x+i y\) in \(D .\) (Hint: If \(z_{0}=x_{0}+i y_{0}\) is in \(D\), let \(0<\left|z-z_{0}\right| \rightarrow 0\), in the definition of \(f^{\prime}\) \(\left(z_{0}\right)\), through \(z\) of the form \(z=x+i y_{0}\), and then of the form \(z=x_{0}+i y\), to obtain $$ \begin{aligned} f^{\prime}\left(z_{0}\right) &=\frac{\partial u}{\partial x}\left(x_{0}, y_{0}\right)+i \frac{\partial v}{\partial x}\left(x_{0}, y_{0}\right) \\ &=\frac{\partial v}{\partial y}\left(x_{0}, y_{0}\right)-i \frac{\partial u}{\partial y}\left(x_{0}, y_{0}\right) \end{aligned} $$ The equations \(\left({ }^{*}\right)\) are called the Cauchy-Riemann equations.)

Step by step solution

01

Understanding the Function

The function \( f \) is a complex-valued function defined on the disk \( D \). It is given that \( f \) is differentiable within this disk. The real part of \( f \), denoted as \( u(x, y) \), is represented by the real part of \( f(z) \) when \( z = x + iy \), and the imaginary part, \( v(x, y) \), is the imaginary part of \( f(z) \).

02

Formulating the Derivative

The derivative of \( f \) at \( z_0 = x_0 + iy_0 \) is denoted as \( f'(z_0) \). This derivative can be approached by taking the limit through real values and then imaginary values of \( z \).

03

Limit Through Real Values

Let \( z = x + iy_0 \). Then, the differentiable function \( f(z) \) gives:\[ f'(z_0) = \lim_{x \to x_0} \frac{f(x + iy_0) - f(x_0 + iy_0)}{x - x_0} \]This limit results in the expression \( \frac{\partial u}{\partial x}(x_0, y_0) + i \frac{\partial v}{\partial x}(x_0, y_0) \).

04

Limit Through Imaginary Values

Now, consider \( z = x_0 + iy \). Then, the differentiable function \( f(z) \) gives:\[ f'(z_0) = \lim_{y \to y_0} \frac{f(x_0 + iy) - f(x_0 + iy_0)}{y - y_0} \]This limit results in the expression \( \frac{\partial v}{\partial y}(x_0, y_0) - i \frac{\partial u}{\partial y}(x_0, y_0) \).

05

Equating the Derivatives

Since \( f'(z_0) \) is the same limit regardless of whether approached through real or imaginary parts, equate the results from Steps 3 and 4:\[ \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} \]Separate into real and imaginary parts:1. \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)2. \( \frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y} \).

06

Concluding the Cauchy-Riemann Equations

The equations derived in Step 5 are recognized as the Cauchy-Riemann equations. These conditions confirm that \( f \) is differentiable as a complex function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cauchy-Riemann Equations

The Cauchy-Riemann equations are crucial for understanding complex differentiation. They provide the conditions under which a complex function is differentiable. Differentiability in the complex sense is a stronger condition than in the real variable sense. For a function \(f(z)\), defined as \(f(z) = u(x, y) + iv(x, y)\), to be differentiable at a point means that both the real part \(u\) and the imaginary part \(v\) must satisfy the Cauchy-Riemann equations. These are:

• \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)
• \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

These equations ensure the consistency required for a complex derivative to exist. Note that if \(f\) satisfies these equations everywhere in a region, then \(f\) is not just differentiable at a single point, but throughout that region.

Complex Differentiation

Complex differentiation delves into how complex functions change with respect to their complex variable. A function \(f(z)\), where \(z = x + iy\), is differentiable at a point \(z_0\) if the derivative \(f'(z_0)\) exists. For complex differentiation, the concept closely resembles differentiation in real analysis, but with added intricacies due to the function's dependence on two real variables. Complex functions must adhere to the Cauchy-Riemann equations for their derivatives to exist.When analyzing \(f'(z)\), you consider small changes in \(z\). The limit process involves both the real and imaginary parts of \(f\). To find \(f'(z_0)\), you might first examine the change in \(z\) across real values, and then independently through imaginary values. The method implies equating two corresponding expressions to ensure that the complex derivative is consistent, leading back to the revered Cauchy-Riemann equations.

Real and Imaginary Parts of Complex Function

In complex analysis, any complex function \(f(z)\) can be decomposed into its real and imaginary parts. If \(z = x + iy\), then \(f(z) = u(x, y) + iv(x, y)\), where \(u(x, y)\) is the real part and \(v(x, y)\) is the imaginary part. Examining these parts is foundational to understanding the behavior and properties of complex functions, especially regarding their differentiability.

• Real Part: \(u(x, y)\)
• Imaginary Part: \(v(x, y)\)

The interplay between these parts matters significantly. The Cauchy-Riemann equations are derived by relating the partial derivatives of \(u\) and \(v\). The articulation of \(f\) in terms of these components aids in analyzing its derivatives and integrals, which is essential for problems in complex analysis.