Implied Functions

Look at my specific interval, it doesn't include the n*cis(π end.

I didn't include it either:

I'm actually not sure what you mean by the bolded. The range of n*cis(θ in the interval [0,π is just [-n,n) in the real plane and [0,n] in the imaginary plane. Wouldn't the real part imply that sqrt(a) can equal -a, because it's in the range?

I'm assuming you're talking about my bolded part. cis(θ's imaginary part is i*sin(θ, which increases from 0 to i*n at π/2 and then back to 0 at π. Therefore, it includes i*n. I meant [0,i*n] by the way.

The imaginary values are not continuous.

It looks like the function is only discontinuous when y=0 and x<0. If y=0, then the imaginary part disappears entirely. The graph seems to be saying that:

For x<0, lim sqrt(x+i*y) as y approaches -0 = -sqrt|x|

and

For x<0, lim sqrt(x+i*y) when x<0 as y approaches +0 = +sqrt|x|

I really don't see how this is possible, but okay. Let's say it is. There is indeed a discontinuity, and it only occurs at y=0 for x<0. I don't see how this disproves the identity:



Take -1 for example. It's possible to split this up into sqrt(x)*sqrt(y) as long as x*y = -1, which pretty much puts them on "opposite" sides of the discontinuity. The discontinuity didn't seem to change anything. The identity still seems to hold for any -C, where C is a constant.

Because sqrt(-1) = -i too. You could use the property sqrt(-1) = i, but then you also have to solve the equation for sqrt(-1) = -i as well. It is thus easier to just use the i^2 = -1 property. This also usually removes possible square root problems you might run into, such as complex square roots.

What? i is defined to be sqrt(-1). The only way I can possibly see this being possible is:

sqrt(-1) = i

sqrt(-i^4) = i

sqrt(i^4)*sqrt(-1) = i

i^2*sqrt(-1) = i

-sqrt(-1) = i

sqrt(-1) = -i

You already said that this is false, because the property:



doesn't hold in the complex domain.