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. $$w=\frac{z-i}{z+i}$$

Short answer

For the function \( w = \frac{z-i}{z+i} \), the real part, \( u \), is \( \frac{x-1}{x^2 + y^2 + 1} \) and the imaginary part, \( v \), is \( \frac{y-1}{x^2 + y^2 + 1} \).

Step-by-step solution

Substitute the variable

Let the complex number be written as: \( z = x + iy \)

Substitute into the given function

Substitute \( z = x + iy \) into the given function \( w = \frac{z - i}{z + i} \), resulting in: \[ w = \frac{x + iy - i}{x + iy + i} \]

Simplify the denominator

Simplify the denominator by multiplying the numerator and denominator by the conjugate of the denominator: \[ w = \frac{(x + iy - i)(x + iy - i)}{(x + iy + i)(x - iy - i)} \] This simplifies to: \[ w = \frac{(x-1) + i(y-1)}{x^2 + 1 + y^2 - 1} = \frac{(x-1) + i(y-1)}{x^2 + y^2 + 1} \]

Separate real and imaginary parts

Separate the real and imaginary parts: \( u(x,y)= \frac{x-1}{x^2 + y^2 + 1} \), \( v(x,y) = \frac{y-1}{x^2 + y^2 + 1} \)

Sketch the graphs

Sketch the graphs in the \( (x, y) \) plane for several values of \( u \) and \( v \). Ensure that the curves \( u = const. \) should be orthogonal to the curves \( v = const. \). This visualization will help to understand the relationship between the real and imaginary parts of the function.

Key concepts

complex number

A complex number is a number that combines a real part and an imaginary part. It is written in the form:

• $$z = x + iy$$

where:

• x is the real part
• y is the imaginary part

The imaginary unit is denoted as $$i$$, and is defined by the property that $$i^2 = -1$$.
In our exercise, we started by expressing the complex number in this form before substituting it into the given function.

real part

The real part of a complex number is the 'x' component in the expression $$z = x + iy$$.
In the exercise, once the function $$w = \frac{z - i}{z + i}$$ was simplified, we extracted the real part from the result.
The real part (u) was given as: $$u(x,y) = \frac{x-1}{x^2 + y^2 + 1}$$. This represents the real component of the function, and it varies based on the inputs x and y. When sketching curves where $$ u = const. $$, we see how this real value moves according to positions on the complex plane.

imaginary part

The imaginary part of a complex number is the 'y' component in the expression $$z = x + iy$$.
After simplifying the function $$w = \frac{z - i}{z + i}$$ in the exercise, we also determined the imaginary part (v) from the result.
It was found as: $$v(x,y) = \frac{y-1}{x^2 + y^2 + 1}$$. This represents the imaginary component of the function, which changes as x and y vary. By sketching curves where $$ v = const. $$, we visualize how the imaginary value takes shape on the complex plane.

orthogonal curves

Orthogonal curves are curves that intersect at right angles (90 degrees).
In the context of the exercise, the curves represented by $$ u = const. $$ and $$ v = const. $$ are orthogonal to each other.
The orthogonality property is important in complex functions, particularly when mapping and understanding function behaviors.
To illustrate this orthogonality in the complex plane, we sketched graphs for several values of $$u$$ and $$v$$ showing how these curves meet perpendicularly.

conformal mapping

Conformal mapping is a function that preserves angles locally.
In other words, it is a transformation that maintains the angles or shapes of small structures.
In our exercise, the function $$w = \frac{z - i}{z + i}$$ is an example of a conformal map.
This type of mapping is used to study complex functions and helps visualize how complex structures behave and interact.
Through conformal mapping, it is ensured that the geometrical properties related to angles are preserved, making them highly useful in fields like fluid dynamics and electromagnetism.