0

i would try to understand meaning of square root function in complex space,square root function defined by

$f(x)=x^{1/2}$

is a very interesting function,let us define it first of all in real numbers,we know that square root from given non negative number is a non negative number,square of which is equal to given number,let say $\sqrt{4}=2$,but we also know that $(-2)^2=4$,it should be considered as square root function is a function where one point is related to one point,or it is one-one,in this case $4$ is converted to $2$,but is's inverse square function is function,on which two point is related to one point,or two-one,in this case $-2$ and $2$ is related to square function,now what about complex space?in many books there was mentioned Riemann space related to square root functions in complex analysis books,please help me to determine why it is mentioned?for example as i know Riemann said that square root function should not be from $C$ to $C$ , but should be defind on the Riemann surface.thanks in advance

dato datuashvili
  • 9,194

1 Answers1

1

Your question is very unclear, but... assuming you are asking about Riemann surfaces, it is possible to concoct a "Riemann surface" $S$ on which the square root function $f(z)$ considered as a mapping from $S$ to $\mathbb{C}$ is one-to-one and onto. I really like the pictures here, and I will not attempt to formalize the definition of Riemann surface here. Just pictures.

I would study these two:

http://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-fall-1999/study-materials/riemann-surfaces-the-logarithm/

http://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-fall-1999/study-materials/riemann-surfaces-the-square-root/

I think it is nice to think of $S$ in this case as two copies of $\mathbb{C}$ where you cut out the imaginary axis and glue them together. Given these two sheets, let's call them $C_1,C_2$, you glue the part above the cut of $C_1$ to the part below the cut of $C_2$ (and top of cut of $C_2$ to bottom of cut of $C_1$). To the first copy $C_1$, associate every $z$ to polar form $z=re^{i\theta}$ but forcing $-\pi < \theta < \pi$. To the bottom, do the same, but force $\pi < \theta < 2\pi$.

In both cases, define $\sqrt{z}$ by halving the angle and square rooting the magnitude: $\sqrt{re^{i\theta}} = \sqrt{r} e^{i\theta/2}$. You'll notice $\sqrt{z}$ on $C_1$ differs from $\sqrt{z}$ on$C_2$ by exactly a negative sign. We've just defined two separate branches of $\sqrt{z}$ on the same cut ($\mathbb{C}$ without imaginary axis). Also you can justify that across the cuts we glued together, this map is continuous (even analytic! Morera's theorem, ish). Now if you consider our newly constructed taped together surface $S$, you can define other branches by making different cuts of $S$ that correspond to a single regions of $\mathbb{C}$ (in general, the largest cut you will get away with will be $\mathbb{C}$ cutting away some path connecting $0$ and $\infty$ (branch points)).

Moreover, you can also think of defining $\sqrt{z}$ on the Riemann surface corresponding to the logarithm (infinitely many copies of $\mathbb{C}$ glued together), and it will no longer be one-to-one on that surface (but it will be periodic, in some sense! so you can consider cutting the logarithm Riemann surface and gluing the ends together in a suitable fashion).

I tried to find a nice illustration of this procedure on google because it has to exist but couldn't.. Maybe someone else can provide.

Evan
  • 3,861