Difference between revisions of "Manuals/calci/IMLOG"

Revision as of 16:54, 2 August 2017

IMLOG (ComplexNumber,Base)

This function shows the log value of a complex number.
In $IMLOG(ComplexNumber,Base)$ , $ComplexNumber$ is any complex number.
$Base$ is the base value of a Log values.
A complex logarithm function is an "inverse" of the complex exponential function.
It is same as the real natural logarithm ln x is the inverse of the real exponential function.
Thus, a logarithm of a complex number z is a complex number w such that $e^{w}=z$ .
The notation for such a $w$ is $lnz$ or $logz$ .
If $z=re^{i\theta }$ with $r>0$ which is in Polar form, then $w=lnr+i\theta$ is one logarithm of z.
Adding integer multiples of 2πi gives all the others.
The complex exponential function is not injective, because $e^{w+2\pi i}=e^{w}$ for any w, since adding iθ to w has the effect of rotating $e^{w}$ counterclockwise θ radians.
So the points $.....w-4\pi i,w-2\pi i,w,w+2\pi i,w+4\pi i....$

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 *Thus, a logarithm of a complex number z is a complex number w such that <math>e^w = z</math>.
 *The notation for such a <math>w</math> is <math>ln z</math> or <math>log z</math>.
-*If <math>z = re^{iθ}</math> with <math>r > 0</math>which is in Polar form, then <math>w = ln r + iθ</math> is one logarithm of z.
+*If <math>z = re^{i\theta}</math> with <math>r > 0</math>which is in Polar form, then <math>w = ln r + i\theta</math> is one logarithm of z.
 *Adding integer multiples of 2πi gives all the others.
 *The complex exponential function is not injective, because <math>e^{w+2\pi i} = e^w</math> for any w, since adding iθ to w has the effect of rotating <math>e^w</math> counterclockwise θ radians.
 *So the points <math>.....w-4\pi i,w-2 \pi i, w, w+2\pi i,w+4 \pi i....</math>