The Complex Logarithm Function | Principal value of the Logarithm | Complex Analysis #5

The video determines the logarithm of a complex number and contains concepts as: The Complex Logarithm function (Multivalued Logarithm), Principal Value of the Logarithm, Argument of a complex number and the Principal Argument of a complex number.

LINK TO COMPLEX ANALYSIS PLAYLIST
   • The Complete Guide to Complex Analysi...  

SUPPORT
Consider subscribing and liking if you enjoyed this video or if it helped you understand the subject. It really helps me a lot.

CONCEPTS FROM THE VIDEO
► The Complex Logarithm function: ln(z) = ln|z| + i*arg(z) = ln|z| + i*(Arg(z) + n*2π) n = 0, +-1, +-2, ...
Is also called the multivalued logarithm ln(z) and is the "inverse" (but not really) of the complex exponential function. For a function to have an inverse, it must map distinct values to distinct values, i.e., be injective. But the complex exponential function is not injective since there exists an infinite number of different angles for the same complex number (e^(z+2πi) = e^z for any z).

► Principal Value of the Logarithm: Ln(z) = ln|z| + i*Arg(z)
Is a consequence of when the n in the complex logarithm function is equal to 0. The principal value Ln(z) is when the angle for the complex number lies in the interval (−π,π].

► Argument of a complex number: arg(z)
Is the angle from the positive real axis to the line joining the point to the origin. One complex number can have an infinite number of angles.

► Principal Argument of a complex number: Arg(z)
Is the angle from the positive real axis to the line joining the point to the origin, but the angle most lie in the interval (−π,π].

TIMESTAMPS
The Complex logarithm: 00:00 - 01:25
Argument of an complex number: 01:25 - 01:49
Principal argument of a complex number: 01:49 - 02:45
The multivalued logarithm: 02:45 - 03:05
Principal Value of the logarithm: 03:05 - 03:40
Example: 03:40 - 04:49
Outro: 04:49 - 05:05

SOCIAL
► Follow me on Youtube: http://bit.ly/1NQhPJ9
► Follow me on Twitter:   / the_mathcoach  

#TheMathCoach #ComplexAnalysis