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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle w} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ln z} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log z} .
• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = re^{i\theta}} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r > 0} which is in Polar form, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w = ln r + i\theta} is one logarithm of z.
• Adding integer multiples of 2πi gives all the others.
• The complex exponential function is not injective, because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{w+2\pi i} = e^w} for any w, since adding iθ to w has the effect of rotating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^w} counterclockwise θ radians.
• So the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle .....w-4\pi i,w-2 \pi i, w, w+2\pi i,w+4 \pi i....}