Difference between revisions of "Manuals/calci/IMLOG"

Revision as of 18:00, 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 $.....w-4\pi i,w-2\pi i,w,w+2\pi i,w+4\pi i....$

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

@@ Line 11: / Line 11: @@
 *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^{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.
+*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>
+==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
+==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 ]]