Chapter 5: Problem 35
This problem is to refresh your memory about some properties of complex numbers needed at several points in this chapter, but especially in deriving the resonance formula (5.64). (a) Prove that any complex number \(z=x+i y\) (with \(x\) and \(y\) real) can be written as \(z=r e^{i \theta}\) where \(r\) and \(\theta\) are the polar coordinates of \(z\) in the complex plane. (Remember Euler's formula.) (b) Prove that the absolute value of \(z\), defined as \(|z|=r,\) is also given by \(|z|^{2}=z z^{*},\) where \(z^{*}\) denotes the complex conjugate of \(z\) defined as \(z^{*}=x-\)iy. \(\left(\text { c) Prove that } z^{*}=r e^{-i \theta} . \text { (d) Prove that }(z w)^{*}=z^{*} w^{*} \text { and that }(1 / z)^{*}=1 / z^{*}\right.\) (e) Deduce that if \(z=a /(b+i c),\) with \(a, b,\) and \(c\) real, then \(|z|^{2}=a^{2} /\left(b^{2}+c^{2}\right)\).
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.