MATH3401
Complex Analysis
Lecture 3
Complex conjugate
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\(\conj{\cdot} : \C \ra \C\), \(x+i\,y \mapsto x-i\,y\). If \(z=x+i\,y \), then \[\overline{z} = x-i\,y.\] |
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Complex conjugate
Properties
(i) \(z = \conj{z} \leftrightarrow \Im(z)=0\), i. e. \(z\in \mathbb R\).
(ii) \(\conj{\left( \conj{z} \right)} = z \).
(iii) \(\conj{\left( z \cdot w \right)} = \conj{z} \cdot \conj{w} \).
(iv) \(\conj{\left( \dfrac{1}{z} \right)} = \dfrac{1}{\conj{z}}, \; z\neq 0 \).
Complex conjugate
Properties
(v) \( |z|^2 = z \cdot \conj{z}\).
(vi) \( \Re(z) = \dfrac{z+\conj{z}}{2}\), \(\Im(z) = \dfrac{z-\conj{z}}{2i}\).
(vii) \(\conj{z + w} = \conj{z} +\conj{w} \).
Triangle inequality
\(|z+w|\leq |z| + |w|\)
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Proof: \(|z+w|^2=|z|^2+|w|^2-2|z||w|\cos A\) \(|z+w|^2\leq |z|^2+|w|^2+2|z||w|\), Then \(|z+w|^2 \leq \left(|z|+|w|\right)^2\). Take square root of both sides and we are done \(\blacksquare\). |
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Polar coordinates
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\( \left\{ \begin{array}{r} x = r\cos \theta \\ y = r\sin \theta \end{array} \right. \) \[ \begin{align*} z &= r e^{i \theta} \\ &= r \left(\cos \theta + i \, \sin \theta \right)\\ &= r \,\text{c}i\text{s}\,\theta \end{align*} \] |
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Argument
\(r e^{i \theta} \) formally follows from Taylor series of \(e^{i\theta}\).
Here \(\theta\) is an argument of \(z\) and we write \(\theta = \arg(z)\).
The \(\arg \) is not a (single-valued) function.
If \(\theta\) is an argument of \(z\), \(\Arg(z)\) is defined to be the unique value of \(\theta\) with \[-\pi \lt \arg (z) \leq \pi\]
Argument
\(\Arg(1+i)=\frac{\pi}{4}\), but \[\arg(1+i) = \ldots, -\frac{7\pi}{4}, \frac{\pi}{4}, \frac{9\pi}{4}, \ldots.\]
\(\Arg(-6)=\pi\).
\(\Arg(0)\) is undefined.
\(\arg(0)=\R\).
Argument
So \(\Arg\) is a function: \[\C\setminus \{0\} \ra (-\pi, \pi] \]
\(\C\setminus \{0\} = \C^* = \C_*\).
Argument
Notice also that \(|e^{i\theta}|=1\) 📝 (← This symbol means 'check it').
\(\left(e^{i\theta} \right)^{-1} = e^{-i\theta}= \conj{e^{i\theta}}\).
\(\left( r e^{i\theta} \right) \left( \rho e^{i\phi} \right) = r \rho e^{i\left(\theta + \phi\right)} \).
So \( |z\, w| = |z|\,|w|\) and \[\arg(z\, w) = \arg(z) + \arg(w).\]
Argument
But \( \Arg(z\, w)\) may fail to be equal to \[\Arg(z)+\Arg(w).\]
E. g. \( z = w = \dfrac{-1+i}{2}\).
\( \Arg(z) = \Arg(w) = \dfrac{3\pi}{4}\).
\( z\,w = -i \implies \Arg(z \, w) = -\dfrac{\pi}{2}\).
but \( \Arg(z)+\Arg(w) = \dfrac{3\pi}{2}\).
de Moivre's formula
Let \(z = r e^{i\theta}\), then \(z^n = r^n e^{i n \theta}\), \(n\in \N\).
For \(r=1\), \[ \begin{align*} e^{i n \theta} &= \left(\cos \theta + \sin \theta \right)^n \\ &= \cos \left(n\theta\right) + i\,\sin \left(n\theta\right) \end{align*} \] de Moivre's formula.
Example: \[ \begin{align*} \left(1+i\right)^7 &= \left( \sqrt{2} e^{i\pi/4} \right)^7 \\ &= \left(\sqrt{2}\right)^7 e^{7\pi i/4} = 8-8i. \end{align*} \]