What is the Difference Between Log, ln, arg, and Arg in Complex Analysis?

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Discussion Overview

The discussion focuses on the differences between the terms log, ln, arg, and Arg in the context of complex analysis. Participants explore definitions, usage in textbooks, and the implications of these terms in mathematical expressions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that their textbooks use log to denote the 'old' logarithm for real values and Log for the complex logarithm.
  • One participant suggests that log(z) = ln|z| + i*arg(z) and Log(z) = ln|z| + i*Arg(z), indicating a distinction between the multivalued and single-valued functions.
  • Another participant questions how ln(2) can be replaced with log(2) in a specific example, suggesting a potential inconsistency in the textbook.
  • There is a mention of the use of n2π (where n is an integer) in relation to the periodic nature of the tangent function, raising questions about its necessity in the context of complex logarithms.
  • Some participants propose that Arg denotes a principal branch and is a single-valued function, while arg is the multivalued version.

Areas of Agreement / Disagreement

Participants express varying interpretations of the terms log, ln, arg, and Arg, with no consensus reached on their definitions or the specific example discussed.

Contextual Notes

There are unresolved questions regarding the definitions of arg and Arg, as well as the implications of replacing ln with log in specific calculations. The discussion reflects different textbook conventions and potential ambiguities in terminology.

sweetvirgogirl
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my complex analysis book uses all three of them...

although i know the difference between log and ln, I'm kinda clueless about Log ... any ideas?
 
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It's probably the principle value of log. I'm sure your book has it mentioned somewhere.
 
sweetvirgogirl said:
my complex analysis book uses all three of them...

although i know the difference between log and ln, I'm kinda clueless about Log ... any ideas?
In my textbook, log was used to denote the 'old' logarithm for real values only, and Log for the complex logarithm (i.e. Log(z) = log(r) + i*phi, with phi the phase, if I recall correctly).
 
TD said:
In my textbook, log was used to denote the 'old' logarithm for real values only, and Log for the complex logarithm (i.e. Log(z) = log(r) + i*phi, with phi the phase, if I recall correctly).
let me tell you how my textbook defines certain terms ...

log z = ln |z| + i * arg z
Log z = ln |z| + i * Arg z

now consider this example:
log (1+ i * 3^(1/2)
now the value I would get is ... ln |1+ i * 3 ^(1/2)| + i ( pi/3 + 2 n pi)
which simplifies to ln 2 + i (pi/3 +2 n pi)

now the answer in the back of the book is log 2 + i (pi/3 +2 n pi)
and this is not the first time they have done it ... so i don't think it's a typo ...

lol ... mind explaning how they replaced ln 2 with log 2??
thanks!
 
another thing ...
why do they always use n2pi (n = 0, , 1, -1, 2, -2...)??
coz the values in case of tangent give the same value for n * pi
 
hmmm... bump?
 
sweetvirgogirl said:
log z = ln |z| + i * arg z
Log z = ln |z| + i * Arg z

Your task is to now look up how they define arg and Arg. Most likely, Arg will denote some 'principle branch' and be a single valued function while arg is the multivalued version.

sweetvirgogirl said:
... mind explaning how they replaced ln 2 with log 2??

In advanced textbooks (=beyond intro calculus) "log" usually denotes the base e logarithm, i.e. the "ln" on your calculator. They probably explain this somewhere in your text.
 

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