96 lines
1.8 KiB
TeX
96 lines
1.8 KiB
TeX
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\section{Complex numbers}
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When Eigenmath starts up, it defines the symbol $i$ as $i=\sqrt{-1}$.
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Other than that, there is nothing special about $i$.
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It is just a regular symbol that can be redefined and used for some other purpose if need be.
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\medskip
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\noindent
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Complex quantities can be entered in rectangular or polar form.
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\medskip
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\verb$a+i*b$
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$$a+ib$$
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\verb$exp(i*pi/3)$
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$$\exp({1\over3}i\pi)$$
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\medskip
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\noindent
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Converting to rectangular or polar coordinates simplifies mixed forms.
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\medskip
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\verb$A=1+i$
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\verb$B=sqrt(2)*exp(i*pi/4)$
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\verb$A-B$
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$$1+i-2^{1/2}\exp({1\over4}i\pi)$$
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\verb$rect$
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$$0$$
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\medskip
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\noindent
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Rectangular complex quantities, when raised to a power, are multiplied out.
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\medskip
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\verb$(a+i*b)^2$
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$$a^2-b^2+2iab$$
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\medskip
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\noindent
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When $a$ and $b$ are numerical and the power is negative, the evaluation is done as follows.
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$$(a+ib)^{-n}=\left[{a-ib\over(a+ib)(a-ib)}\right]^n=\left[{a-ib\over a^2+b^2}\right]^n$$
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Of course, this causes $i$ to be removed from the denominator.
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%For $n=1$ we have
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%$${1\over a+ib}={a-ib\over a^2+b^2}$$
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Here are a few examples.
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\medskip
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\verb$1/(2-i)$
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$${2\over5}+{1\over5}i$$
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\verb$(-1+3i)/(2-i)$
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$$-1+i$$
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\newpage
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\noindent
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The absolute value of a complex number returns its magnitude.
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\medskip
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\verb$abs(3+4*i)$
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$$5$$
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\medskip
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\noindent
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In light of this, the following result might be unexpected.
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\medskip
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\verb$abs(a+b*i)$
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$$\mathop{\rm abs}(a+ib)$$
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\medskip
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\noindent
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The result is not $\sqrt{a^2+b^2}$ because that would assume that
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$a$ and $b$ are real.
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For example, suppose that $a=0$ and $b=i$.
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Then
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$$|a+ib|=|-1|=1$$
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and
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$$\sqrt{a^2+b^2}=\sqrt{-1}=i$$
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Hence
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$$|a+ib|\ne\sqrt{a^2+b^2}\quad\hbox{for some $a,b\in\cal C$}$$
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\medskip
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\noindent
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The {\it mag} function is an alternative.
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It treats symbols like $a$ and $b$ as real.
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\medskip
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\verb$mag(a+b*i)$
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$$(a^2+b^2)^{1/2}$$
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