A Even with a whimsical mathematical usage, solutions are obtained!

AI Thread Summary
The discussion highlights the coherence of solutions derived from whimsical applications of real logarithm properties to complex logarithms, emphasizing that these properties hold true for analytic functions. It asserts that a real function is essentially a restriction of its complex counterpart, and two analytic functions can only be identical on the real line if they are identical overall. The term "analytical" is explored, noting its historical association with functions expressible as series. The conversation also touches on the equivalence of "analytical" and "holomorphic" in complex analysis, referencing the Cauchy-Riemann equations as a foundational definition. Overall, the thread underscores the interconnectedness of real and complex analysis in mathematical functions.
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Even with a whimsical mathematical usage, coherent solutions are obtained!
Hello everyone,
logcomplexe 1.JPG

logcomplexe 2.JPG

Here, we observe that the familiar properties of the real logarithm hold true for the complex logarithm in these examples.

So why does a whimsical mathematical use of real logarithm properties yield coherent solutions even in the case of complex logarithm?
 

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As long as you are dealing with analytic functions, the real function is just a restriction of the complex analytic function. Two analytic functions can only be identical on the real line if they are identical.
 
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FactChecker said:
As long as you are dealing with analytic functions, the real function is just a restriction of the complex analytic function. Two analytic functions can only be identical on the real line if they are identical.
The term analytical is very interesting. It is reserved for functions that have an expression as a series. And that was how mathematicians regarded all functions for a long time, as series.
 
fresh_42 said:
The term analytical is very interesting. It is reserved for functions that have an expression as a series. And that was how mathematicians regarded all functions for a long time, as series.
When I studied complex analysis, "analytical" and "holomorphic" were assumed to mean more or less the same thing.
 
Svein said:
When I studied complex analysis, "analytical" and "holomorphic" were assumed to mean more or less the same thing.
Isn't that still the case?
 
fresh_42 said:
Isn't that still the case?
I certainly hope so. But the definition used to be "functions that satisfy the Cauchy-Riemann equations".
 
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