Exploring the Result of a^b with a Real or Complex Number

[JPC]

hey, was wondering what would happen if i do :

a^b
with a : real or complex number
and b : a complex number

like : 2^i

 

[genneth]

The definition:

$$a^b = \exp (b \ln a)$$

In general ln is multivalued (actually infinitely valued) so there is no unique answer.

 

[JPC]

whats exp() ?
i know log() , and ln() but never heard of exp()

 

[edmondng]

exponential
e = 2.718281828

 

[genneth]

Specifically, $$exp(x) = e^x$$.

 

[JPC]

oh ok :

so
2^i = e^ ( i * ln(2))
?

 

[Gib Z]

Yup that is perfectly correct. Or if you want it split into real and imaginary parts: $$\cos (\ln 2) + i \sin (\ln 2)$$

 

[JPC]

Yup that is perfectly correct. Or if you want it split into real and imaginary parts: $$\cos (\ln 2) + i \sin (\ln 2)$$

ok
now that we started working on exp(x) in my class i know a bit more what you are talking about

But how do u pass from e^(i * ln(2) ) to $$\cos (\ln 2) + i \sin (\ln 2)$$ ?

 

[SiddharthM]

remember that exp(x) and ln(x) are defined in a more general sense as complex functions.
if a complex number is of the form z=a +ib then
exp(z)=exp(a+ ib)=exp(a)exp(ib) by additivity of exponentials and then using euler's formula: exp(z)=exp(a)(cosb +isin(b))
here the a is real so exp(a) is evaluating using the real definition.

The complex definitions of the trancesendental functions extend that of it's real analogue.

JPC: if you learning about the real exponential function for the first time you probably won't encounter euler's formula until later in the course - it's an application of power series.

 

[Gib Z]

JPC, it seems my assumptions on your maths education are wrong, in Australia for some reason we do complex numbers after Exponential functions, forgive the pun but here they are seen as, well, more complex. Anyway, You obviously don't need to know what I say from here, so you can either choose to ignore my post completely or read more on what I say next: There is a famous relation that Euler derived through Taylor Series expansions that told him that $$e^{ix} = \cos x + i \sin x$$. I Merely used that identity straight off.

 

[bchui]

For $$b=x+iy$$ , $$a^b=a^{x+ i y}=a^x a^{i y} = a^x \exp(i y \ln (a)) = a^x (\cos (y \ln (a))+i \sin (y \ln (a))$$
by Euler's formula $$e^{i\theta}=\cos\theta+i\sin\theta$$.

I wonder what happens to $$A^B$$ where both $$A$$ and $$B$$ are matrices