Imaginary to the power of fractional number

Question

I am sorry if it's stupid, but I don't know how to calculate imaginary number "$i$" power to fractional number. I know:

$$\begin{aligned} i^2 &= -1\\ i^3 &= \;\;\;i \\ i^4 &= -1 \end{aligned}$$

and so on. I know that multiplying a complex number by "$i^n$" rotates the magnitude vector of that complex number, like: $i(a+bi)$ rotates for $90$ degrees, $i^2(a+bi)$ rotates it for $180$ degrees, to the power of $3$ rotates for $270$ degrees... and so on. And I can even know how to calculate it:

$$\begin{aligned} i(a+bi) &= -b + ai\\ i^2(a+bi) &= -a + bi \end{aligned}$$

and so on. But my problem is, I know multiplying a complex number by $i^{0.5}$ rotates for $45$ degrees, but I have no idea how to calculate it, and represent as complex:

$$ i^{0.5} (a + bi) = ????? $$

Could anyone explain it in simply words for simply guy like me? Or give some link with explanation? Thanks in advance for any help.

PS. Of course I know it's something like:

$$ i^{0.5}(a + bi) = (0.707\ldots * a) + (0.707\ldots * b)i $$

But have no idea how it happened :)

Answer 1

You are right, is not as easy than a power by an integer. You must to use the De Moivre's formula

Answer 2

What you do is first choose a branch of the logarithm, for now take the principal branch. Then define $$z^x=e^{x\log(z)}$$ Notice that this works for any $x\in\mathbb{C} $, not just fractions.

Answer 3

(Slight correction in your question: $i^3 = -i$, and $i^4 = i^0 = 1$.)

Maybe polar representation will help.

If you have a complex number, you can express it either as real and imaginary parts $(a+bi)$ or as magnitude and angle in the complex plane ($re^{i\theta}$, with $r$ being how long it is, and $\theta$ being what angle it makes with the positive real (positive $x$) axis).

So your number is either represented as $0 + 1i$, or $1e^{i\pi/2}$.

Taking the square root of the latter expression directly give you one value: $\sqrt{1}e^{i\pi/4}$. (There's another value, $1e^{i5\pi/4}$, which comes by applying De Moivre's theorem.) So this square root of $i$ makes an angle of $\pi/4$ (or $45^{\circ}$) in the complex plane, and has a length of $1$. (The other is in the third quadrant.)

In real-imaginary form, this is $\sqrt{2}/2 + i\sqrt{2}/2$, which is what you wrote at the end. You can convert from polar to this form by the formula $r\cos \theta + i r \sin \theta$.

You can then use this expression to get a value for $i^{0.5}(a+bi)$ with the FOIL method.

Hope this helps!

Answer 4

did a bit of math, correct me if I'm wrong, but I got the following:

$$(x+yi)^2 = \left(\sqrt{x^2+y^2}\right)^n \cdot ( \cos(n\arctan(y/x) ) + i\sin(n\arctan(y/x)) )$$

I got this by converting the complex number to polar coordinates, factoring out the radius and using De Moivre's formula to get the exponentiated complex part. I am not using the proper terminology because frankly I just learned that polar complex numbers are a thing.