Problem 14
Question
\(u=\frac{x^{3}+x y^{2}+x}{x^{2}+y^{2}}, v=\frac{x^{2} y+y^{3}-y}{x^{2}+y^{2}} ; \quad \frac{\partial u}{\partial x}=\frac{x^{4}+2 x^{2} y^{2}-x^{2}+y^{2}+y^{4}}{\left(x^{2}+y^{2}\right)^{2}}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=\frac{-2 x y}{\left(x^{2}+y^{2}\right)^{2}}=-\frac{\partial v}{\partial x}\) \(f\) is analytic in any domain not containing \(z=0\)
Step-by-Step Solution
Verified Answer
The function \( f=u+iv \) is analytic in any domain that does not include \( z=0 \).
1Step 1: Understand the Problem Statement
We are given two functions \( u \) and \( v \) which are expressed as quotients of polynomials in \( x \) and \( y \). We need to verify if these expressions satisfy the conditions of the Cauchy-Riemann equations for the function \( f \) to be analytic.
2Step 2: Identify Cauchy-Riemann Equations
The Cauchy-Riemann equations for the function \( f=u+iv \) are: \( \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} \). These ensure that the function \( f \) is analytic.
3Step 3: Verify Given Partial Derivatives
We are provided with \( \frac{\partial u}{\partial x} = \frac{x^4 + 2x^2y^2 - x^2 + y^2 + y^4}{(x^2 + y^2)^2} \) and \( \frac{\partial v}{\partial y} = \frac{x^4 + 2x^2y^2 - x^2 + y^2 + y^4}{(x^2 + y^2)^2} \), which shows \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \). Similarly, \( \frac{\partial u}{\partial y} = \frac{-2xy}{(x^2 + y^2)^2} \) and \( -\frac{\partial v}{\partial x} = \frac{-2xy}{(x^2 + y^2)^2} \) confirm \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
4Step 4: Confirm Analyticity
Since both pairs of Cauchy-Riemann equations are satisfied by the partial derivatives given, function \( f=u+iv \) is analytic. It's noted that \( f \) is analytic in any domain that does not include \( z=0 \), which means \( x^2 + y^2 eq 0 \) to avoid division by zero in the expressions.
Key Concepts
Partial DerivativesAnalytic FunctionsComplex Analysis
Partial Derivatives
Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of several variables. They measure how a function changes as one of its input variables changes, while keeping the other variables constant. In other words, a partial derivative is the derivative of a function with respect to one variable, with all others treated as constants. This is essential when dealing with complex-valued functions of real variables, like the ones in complex analysis.
For the given functions \( u \) and \( v \), calculating partial derivatives reveals important relationships. We compute \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \), as well as \( \frac{\partial v}{\partial x} \) and \( \frac{\partial v}{\partial y} \). These derivatives are then compared to verify certain conditions. By understanding how each variable individually impacts the function, one can assess whether the function maintains a property known as analyticity, which is crucial for further studies involving the Cauchy-Riemann equations.
For the given functions \( u \) and \( v \), calculating partial derivatives reveals important relationships. We compute \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \), as well as \( \frac{\partial v}{\partial x} \) and \( \frac{\partial v}{\partial y} \). These derivatives are then compared to verify certain conditions. By understanding how each variable individually impacts the function, one can assess whether the function maintains a property known as analyticity, which is crucial for further studies involving the Cauchy-Riemann equations.
Analytic Functions
Analytic functions are special kinds of functions in complex analysis with derivatives at every point in their domain. For a function to be considered analytic, it must satisfy the Cauchy-Riemann equations. Analyticity allows the function to be expressed as a power series, similar to a Taylor series but for complex variables.
In this exercise, the goal is to show that the function \( f = u + iv \) is analytic in a domain excluding the point \( z = 0 \). To be considered analytic, the function must satisfy the Cauchy-Riemann equations, represented as \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). Given we have verified these conditions, the function is analytic in any domain that does not include the point where the denominator is zero.
In this exercise, the goal is to show that the function \( f = u + iv \) is analytic in a domain excluding the point \( z = 0 \). To be considered analytic, the function must satisfy the Cauchy-Riemann equations, represented as \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \). Given we have verified these conditions, the function is analytic in any domain that does not include the point where the denominator is zero.
Complex Analysis
Complex analysis is a branch of mathematics that studies functions of complex numbers, blending algebra and calculus into a beautiful theory. It extends many concepts from real analysis into the complex plane, providing powerful tools for solving problems that seem challenging in real-only frameworks.
A key component in complex analysis involves understanding functions like \( f = u + iv \), where \( u \) and \( v \) are real-valued functions of real variables \( x \) and \( y \). These are often decomposed into real and imaginary parts. When these functions satisfy the Cauchy-Riemann equations, they are not just any functions—they have special properties, like being infinitely differentiable and representable by a power series. This analytical framework provides insight into various mathematical phenomena and applications, from fluid dynamics to electrical engineering and beyond. Understanding these concepts helps students grasp how structures identified as analytic functions behave and what implications that behavior has within both theoretical and applied settings.
A key component in complex analysis involves understanding functions like \( f = u + iv \), where \( u \) and \( v \) are real-valued functions of real variables \( x \) and \( y \). These are often decomposed into real and imaginary parts. When these functions satisfy the Cauchy-Riemann equations, they are not just any functions—they have special properties, like being infinitely differentiable and representable by a power series. This analytical framework provides insight into various mathematical phenomena and applications, from fluid dynamics to electrical engineering and beyond. Understanding these concepts helps students grasp how structures identified as analytic functions behave and what implications that behavior has within both theoretical and applied settings.
Other exercises in this chapter
Problem 13
$$e^{-i z}=e^{y-x i}=e^{y} \cos x-i e^{y} \sin x$$
View solution Problem 14
$$\cos \left(\frac{\pi}{2}+i \ln 2\right)=\cos \frac{\pi}{2} \cosh (\ln 2)-i \sin \frac{\pi}{2} \sinh (\ln 2)=-i \cdot \frac{e^{\ln 2}-e^{\ln 2^{-1}}}{2}=-i \cd
View solution Problem 14
$$\frac{i}{1+i} \cdot \frac{1-i}{1-i}=\frac{i+1}{2}=\frac{1}{2}+\frac{1}{2} i$$
View solution Problem 15
\(\frac{e^{i z}-e^{-i z}}{2 i}=2\) gives \(e^{2(i z)}-4 i e^{i z}-1=0 .\) By the quadratic formula, \(e^{i z}=2 i \pm \sqrt{3} i\) and so $$i z=\ln [(2 \pm \sqr
View solution