Taking a number to a complex number power

[ucbugrad]

How do you compute the following?

2^it where t is a real number

while I am at it, how do you compute powers that are not integers

ie: 2^3.14

 

[The1337gamer]

2^(it) = exp(it(log2)) = exp(i(tlog2)) = cos(tlog2) + isin(tlog2)

I guess we would define

2^(3.14) = exp(3.14(log2))

I'm not sure if i could write it in any other way, and i doubt i could compute that in my head but they are equivalent.

Sorry for the bad formatting if it isn't clear.

Edit: These logs are base e.

 

[HallsofIvy]

"3.14", as opposed to $\pi$, is a rational number. It is, in fact, $314/100= 157/50$ so $2^{3.14}= 2^{157/50}= \sqrt[50]{2^{157}}$.

$\pi$ is not rational but there exist a sequence of rational numbers that converge to it (the sequence 3, 3.1, 3.14, ..., for example). $2^\pi$ is equal to the limit of the sequence $2^3$, $2^{3.1}= 2^{31/10}= \sqrt[10]{2^{31}}=$, $2^{3.14}= 2^{314/100}= \sqrt[50]{2^{157}}$,...

Of course, to actually calculate those things you would use $2^a= e^{a ln(2)}$ as The1337gamer said.

 

[ucbugrad]

Thank you all, both addressed all of my issues. Halls of Ivy: where can I find more about this sequence that converges to Pi? What is the formula for each term?

 

[CellCoree]

In order to compute 2it, we can use the complex number formula a+bi, where a and b are real numbers and i is the imaginary unit. In this case, a=0 and b=2t. Therefore, the complex number can be written as 0+2ti. To compute the power of this complex number, we can use the De Moivre's theorem, which states that for any complex number r(cosθ + isinθ), the nth power can be calculated as r^n(cos(nθ) + isin(nθ)). In this case, r=2t and θ=π/2. Therefore, the power of 2it can be computed as (2t)^n(cos(nπ/2) + isin(nπ/2)).

For powers that are not integers, such as 23.14, we can use the same formula. In this case, r=23.14 and θ=0 (since the real part is positive and the imaginary part is 0). Therefore, the power of 23.14 can be computed as (23.14)^n(cos(0) + isin(0)) = (23.14)^n(1+0i) = (23.14)^n.

It is important to note that when computing powers of complex numbers, we must use radians for the angle θ in the De Moivre's theorem. Additionally, the result of the power will also be a complex number, so we must use the appropriate notation a+bi to represent it.