Difference between revisions of "Manuals/calci/IMLOG"
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(Created page with "<div style="font-size:30px">'''IMLOG (ComplexNumber,Base)'''</div><br/> *<math>ComplexNumber</math> is any complex number. *<math>Base</math> is the base value of the Log. ==...") |
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<div style="font-size:30px">'''IMLOG (ComplexNumber,Base)'''</div><br/> | <div style="font-size:30px">'''IMLOG (ComplexNumber,Base)'''</div><br/> | ||
| − | *<math>ComplexNumber</math> is any complex number. | + | *<math>ComplexNumber</math> is any complex number of the form x+iy. |
*<math>Base</math> is the base value of the Log. | *<math>Base</math> is the base value of the Log. | ||
| + | **IMLOG(),returns the logarithm of a complex number to the given base. | ||
==Description== | ==Description== | ||
| Line 11: | Line 12: | ||
*Thus, a logarithm of a complex number z is a complex number w such that <math>e^w = z</math>. | *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>. | *The notation for such a <math>w</math> is <math>ln z</math> or <math>log z</math>. | ||
| − | *If <math>z = re^{ | + | *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. | *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. | *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> | *So the points <math>.....w-4\pi i,w-2 \pi i, w, w+2\pi i,w+4 \pi i....</math> | ||
| + | |||
| + | ==Examples== | ||
| + | # IMLOG("2+3i",2) = 1.850219859070546+ⅈ1.417871630745722 | ||
| + | # IMLOG("9-5i",3) = 2.122422597222964-ⅈ0.4615809504617068 | ||
| + | # IMLOG("9-5i",6) = 1.3013574573492332-ⅈ0.2830170640096076 | ||
| + | # IMLOG("54",5) =2.4784951415313494+ⅈ0 | ||
| + | # IMLOG("-19i",9) = 1.3400719296231876-ⅈ0.7149002168450317 | ||
| + | |||
| + | ==Related Videos== | ||
| + | |||
| + | {{#ev:youtube|v=mO-K8ZCdvfQ|280|center|Complex Logarithm}} | ||
| + | |||
| + | ==See Also== | ||
| + | *[[Manuals/calci/LOG| LOG]] | ||
| + | *[[Manuals/calci/LOGINV| LOGINV]] | ||
| + | *[[Manuals/calci/ANTILOG| ANTILOG]] | ||
| + | |||
| + | ==References== | ||
| + | *[https://en.wikipedia.org/wiki/Complex_logarithm Complex Log] | ||
| + | |||
| + | *[[Z_API_Functions | List of Main Z Functions]] | ||
| + | |||
| + | *[[ Z3 | Z3 home ]] | ||
Latest revision as of 15:06, 22 February 2019
IMLOG (ComplexNumber,Base)
- is any complex number of the form x+iy.
- is the base value of the Log.
- IMLOG(),returns the logarithm of a complex number to the given base.
is any complex number of the form x+iy.
is the base value of the Log.
Description
- This function shows the log value of a complex number.
- In , is any complex number.
- 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 .
- The notation for such a is or .
- If with which is in Polar form, then is one logarithm of z.
- Adding integer multiples of 2πi gives all the others.
- The complex exponential function is not injective, because for any w, since adding iθ to w has the effect of rotating counterclockwise θ radians.
- So the points
,
.
is 
or 
.
with 
which is in Polar form, then 
is one logarithm of z.
for any w, since adding iθ to w has the effect of rotating 
counterclockwise θ radians.
Examples
- IMLOG("2+3i",2) = 1.850219859070546+ⅈ1.417871630745722
- IMLOG("9-5i",3) = 2.122422597222964-ⅈ0.4615809504617068
- IMLOG("9-5i",6) = 1.3013574573492332-ⅈ0.2830170640096076
- IMLOG("54",5) =2.4784951415313494+ⅈ0
- IMLOG("-19i",9) = 1.3400719296231876-ⅈ0.7149002168450317