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2bornot2b
 3 years ago
\[\huge{a^b=e^{b~Loga}}\]
Here a and b are complex numbers.
This is the definition of complex exponent. Why is it defined so?
2bornot2b
 3 years ago
\[\huge{a^b=e^{b~Loga}}\] Here a and b are complex numbers. This is the definition of complex exponent. Why is it defined so?

This Question is Closed

bmp
 3 years ago
Best ResponseYou've already chosen the best response.1Can't we write \( \LARGE e^{b \log{a}}\) as \(\LARGE e^{\log{a} ^ b} = a^b \). Log here is the natural log.

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0@bmp are you sure that you made the above statement keeping in mind that a and b are complex numbers?

estudier
 3 years ago
Best ResponseYou've already chosen the best response.0Isn't it because you are at the same time defining the complex log? In analogy with the usual exp and log functions.

bmp
 3 years ago
Best ResponseYou've already chosen the best response.1Ah, yeah. That's true. My bad. Hmm.

bmp
 3 years ago
Best ResponseYou've already chosen the best response.1I think @estudier is correct. The complex log will be of the form ln(z) = lnz + iArg(z). Because \( \LARGE e^{zw} = e ^{(z + 2\pi i)w}\) and \(\LARGE e ^{(z + 2\pi i)} = ln(z) \)And then I think the equality holds. But I am not sure.

shivam_bhalla
 3 years ago
Best ResponseYou've already chosen the best response.1If you are looking to prove that \[\huge{e^{b~Loga}=a^b}\] Then consider LHS and then \[\huge{e^{Loga^b}}\] Now apply this rule \[\huge a^{\log_b (c)} = c^{\log_b(a)}\] Therefore we get \[\huge{a^b}\] LHS = RHS

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0Is this true ?\[e^{Log~a^b}=e^{b~Log~a}\] Please note we are talking about complex a and b

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0Even if b were an integer, I couldn't have said that, I think.

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0However I could have said that without any hesitation if b was some 1/m where m is an integer

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0Please correct me if I am wrong

shivam_bhalla
 3 years ago
Best ResponseYou've already chosen the best response.1If b were an integer , then this rule would blindly apply. But in the case of complex numbers, I think you are right. We can't say that

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0If a were a complex number and b were an integer, even then we couldn't have said that. Isn't it?

shivam_bhalla
 3 years ago
Best ResponseYou've already chosen the best response.1Have a look at this http://math.fullerton.edu/mathews/c2003/ComplexFunComplexPowerMod.html

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0@cshalvey If you are here can you clarify one doubt, which is the correct statement? 1. "However I could have said that without any hesitation if b \(was\) some 1/m where m is an integer" 2. "However I could have said that without any hesitation if b \(were\) some 1/m where m is an integer"

shivam_bhalla
 3 years ago
Best ResponseYou've already chosen the best response.1@2bornot2b , the b statement is better :D and have a look at the link which I gave above

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0Yes, I am looking at that @shivam_bhalla Thanks for that!

shivam_bhalla
 3 years ago
Best ResponseYou've already chosen the best response.1An example to prove that the statement you gave above in your question is valid is \[\large i^i = e^{i\ln i} = e^{i\ln\left(\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}\right)} = e^{i\ln e^{\frac{i\pi}{2}}} = e^{\frac{\pi}{2}}\]

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0@shivam_bhalla In case you feel that my question is not clear enough let me give you an analogy to explain what exactly I am searching for. If someone asks you why is the complex exponential function defined in the way it's defined you can say that it is done so that the function maintains its property of returning the function on differentiation. Now after knowing this there is some meaning to the definition of complex exponential function. I am searching for a similar thing here. My point is I could have defined it any way, but why in that particular way?

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0take log on both sides, it is equivalent.

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0log or Log? a and b are complex by the way

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0Can you tell me one thing, is the product \(w~log a\) commutative. If so, then I believe the problem is solved.

shivam_bhalla
 3 years ago
Best ResponseYou've already chosen the best response.1@2bornot2b , The answer to your question lies here > http://www.milefoot.com/math/complex/exponentofi.htm It is too long for me to post here

2bornot2b
 3 years ago
Best ResponseYou've already chosen the best response.0And also what is the convention? \[e^{b~Log ~a}=({e^b})^{Log~a}~or~=e^{(b~Log~a)}\]

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0let a be a complex number, then a is equivalent to, \( a = e^{\lna + i \arg(a)}\) \( \ln a = lna + i \arg(a) \) They are equivalent. \( a^b = (e^{\lna + i \arg(a)})^b = e^{b\ln a}\)

shivam_bhalla
 3 years ago
Best ResponseYou've already chosen the best response.1@2bornot2b , Hope you got your answer :D
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