Question

For each of the following functions \(w=f(z)=u+i v,\) find \(u\) and \(v\) as functions of \(x\) and \(y .\) Sketch the graphs in the \((x, y)\) plane of the images of \(u=\) const. and \(v=\) const. for several values of \(u\) and several values of \(v\) as was done for \(w=z^{2}\) in Figure \(9.3 .\) The curves \(u=\) const. should be orthogonal to the curves \(v=\) const. \(w=\sqrt{z} .\) Hint: This is equivalent to \(w^{2}=z ;\) find \(x\) and \(y\) in terms of \(u\) and \(v\) and then solve the pair of equations for \(u\) and \(v\) in terms of \(x\) and \(y\). Note that this is really the same problem as Problem 1 with the \(z\) and \(w\) planes interchanged.

Short answer

Equate and solve the equations: \(u^2 - v^2 = x\) and \(2uv = y\), the lines \(u = \text{const.}\) and \(v = \text{const.}\) are orthogonal.

Step-by-step solution

Express the transformation in terms of real and imaginary parts

Given that the transformation is defined as \(w = f(z) = u + iv\), and \(w = \sqrt{z}\), start by equating \(w^2 = z\). Let \(z = x + iy\) where \(x\) and \(y\) are real.

Substitute and simplify the equation

Substitute \(w = u + iv\) and \(z = x + iy\) into the equation \(w^2 = z\). This gives: \[(u + iv)^2 = x + iy\] Expanding the left-hand side, we have: \[u^2 - v^2 + 2uvi = x + iy\]

Equate real and imaginary parts

From the previous equation, equate the real parts and the imaginary parts separately:For the real part: \[u^2 - v^2 = x\] For the imaginary part: \[2uv = y\]

Solve for \(u\) and \(v\) in terms of \(x\) and \(y\)

Given the equations: \[u^2 - v^2 = x\]\[2uv = y\]solve this system of equations to express \(u\) and \(v\) in terms of \(x\) and \(y\).

Sketch the graph of \(u = \text{const.}\) lines

To find the curves for constant \(u\) values on the \((x, y)\) plane, use the equation: \[u^2 - v^2 = x\] Solve for \(y\) in terms of \(x\) by holding \(u\) constant in this equation.

Sketch the graph of \(v = \text{const.}\) lines

To find the curves for constant \(v\) values on the \((x, y)\) plane, use the equation: \[2uv = y\] Solve for \(x\) in terms of \(y\) by holding \(v\) constant in this equation.

Demonstrate orthogonality

Note that because the Cauchy-Riemann equations hold, the lines \(u = \text{const.}\) and \(v = \text{const.}\) are orthogonal in the \((x, y)\) plane.

Key concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are fundamental in complex function analysis. These equations provide a criterion to check whether a complex function is holomorphic, meaning it has a complex derivative at every point in its domain. Specifically, if a complex function \( f(z) = u(x, y) + iv(x, y) \) is differentiable, the following conditions must be satisfied: \ \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \] \ Here, \( u \) and \( v \) are the real and imaginary parts of the function, respectively. These relations are not just conditions for differentiability; they also ensure the conformal (angle-preserving) nature of the function. In the context of the original exercise where \( w = \sqrt{z} \), verifying the Cauchy-Riemann equations guarantees that the transformation preserves orthogonal grid lines, which is crucial for understanding how curves of constant \( u \) and \( v \) intersect orthogonally in the \( (x, y) \) plane.

Imaginary and Real Parts of Complex Functions

In complex analysis, any complex function \( w = f(z) = u + iv \) can be decomposed into its real part \( u(x, y) \) and imaginary part \( v(x, y) \). This decomposition is vital because it allows us to analyze and solve problems using real functions of real variables. For instance, in our exercise: \ * We start by expressing \( z = x + iy \) and \( w = u + iv \). \ * By substituting into the equation \( w^2 = z \), we separate real and imaginary parts to get real equations: \[ u^2 - v^2 = x \quad \text{and} \quad 2uv = y \] \ These equations link the real components \( (x, y) \) with the real and imaginary parts of \( w \), making it possible to sketch how these parts behave. For instance, lines of constant \( u \) and constant \( v \) in the \( (x, y) \) plane transform according to these real equations. Understanding this relationship is essential for visualizing complex transformations.

Orthogonal Curves in Complex Functions

In complex analysis, curves \( u = \text{const.} \) and \( v = \text{const.} \) are significant due to their perpendicularity, facilitated by the Cauchy-Riemann equations. Orthogonal curves are vital because they simplify visualizing complex transformations. For the function \( w = \sqrt{z} \), let's break it down: \ * From the exercise, we get the equations \( u^2 - v^2 = x \) (real part) and \( 2uv = y \) (imaginary part). \ * Holding \( u \) constant, \( u^2 - v^2 = x \) essentially forms a quadratic relation in the \( (x, y) \) plane, displaying hyperbolic curves. \ * Similarly, for constant \( v \), \( 2uv = y \) gives linear relationships. These relationships show hyperbolic and linear structures that intersect orthogonally. \ Orthogonality confirms the configuration of grid lines remains preserved, and such visualizations help you master the geometric interpretation of complex functions.