Manuals/calci/IMLOG

IMLOG (ComplexNumber,Base)

• $ComplexNumber$ is any complex number.
• $Base$ is the base value of the Log.

Description

• 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 Failed to parse (syntax error): {\displaystyle z = re^{iθ}} with $r>0$which is in Polar form, then Failed to parse (syntax error): {\displaystyle w = ln r + iθ} 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 $.....w-4\pi i,w-2\pi i,w,w+2\pi i,w+4\pi i....$