A Homemade Exponential Equation | Problem 297

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[@XJWill1]

It is not ambiguous. The definition of complex exponentiation is z^w = exp( log(z) * w ) which is multi-valued, and the principal value is z^w = exp( Log(z) * w ) for - pi < Im{ Log(z) }

[@XJWill1]

As for your solution, the denominator in question must be 1 for it to solve the given equation.

[@aplusbi]

So n=0. The others cause multi-valued results and are not valid?

[@FrancisZerbib]

1 = i^4k then compare powers

[@aplusbi]

Good thinking!

[@angeluomo]

I probably did this too simplistically. I just assumed that i, which is the sqrt(-1), equals exactly 1 whenever it is raised to a power that is a multiple of 4 (e.g., i^4 = 1, i^8 = 1, etc.). So to solve the problem, I just worked out when e^z is equal to a multiple of 4 and got e^z= ln(4k)+2i*pi*n. Did I do something wrong here? Just wondering.

[@sarv1494]

Z = ln(4pi) + pi/2i + i2pin, n is a integer

[@FrancisZerbib]

z = log 4k + 2(pi)ni

[@Nobodyman181]

Yes! My answer too!

[@mikecaetano]

Ambiguity is fun!

[@aplusbi]

Absolutely!

[@FisicTrapella]

Some people say that a^(b/c) = (a^b)^(1/c) = (a^(1/c))^b only works (it is well defined) when the base a is greater or equal to zero (a>=0) and it's undefined in the other cases.🤷‍♂

[@XJWill1]

It is not undefined. However, unless a is positive real-valued, the common power identities can fail. But there are more complicated identities that hold for complex-valued a, b, c from your example.

[@FisicTrapella]

@@XJWill1 I just say that for some people, the problem is equivalent to divide by 0 or to the expresion 0/0... So, they say it's undefined (not defined) because it can lead to a different results.