In schaumm's series book for complex variables, I found this question
Why this fallany happens
-1=\sqrt(-1)*\sqrt(-1)=\sqrt(-1*-1)=\sqrt(1)=1
Could someone help me find the error?
Thank you so much.
You are applying multiplication rules that do not apply to complex numbers.
If $\displaystyle z=r\exp(i\theta)~~w=s\exp(i\phi)$ then $\displaystyle z\cdot w=rs\exp(i(\theta+\phi))$.
Now note that $\displaystyle i=\exp\left(\frac{i\pi}{2}\right)$ so $\displaystyle i\cdot i=?$
No, $\displaystyle \sqrt{1}$ is a real number and as such it has only one value $\displaystyle \sqrt{1}=1$.
The complex numbers are enlargement of the real numbers.
The number $\displaystyle i$ is introduced as the root of the equation $\displaystyle x^2+1=0$.
You can see that then $\displaystyle i^2$ must $\displaystyle =-1$.
So it is easy to see why in history it was tempting to say $\displaystyle i=\sqrt{-1}$.
I'm rather confused, Plato.
In the complex system, is \sqrt(1) means we want to find complex numbers z such that z^2=1?
That means in complex system, the \sqrt(1) = +1 and -1, doesn't it?
It differs from the real system that defines \sqrt(1) as a positive number x such that x^2=1. Thus the solution is just x=1, without x=-1?
CMIW
Plato, concerning \sqrt{-4}=\sqrt{4\cdot-1}=\sqrt{4}*\sqrt{-1}=2i.
But is it true when we want to find for example the solution of z^2-i*z+2=0, then we could use
z_12=(i \pm \sqrt((-i)^2-4*1*2))/2=(i \pm \sqrt(-9))/2=(i \pm \sqrt(9*(-1)))/2=(i \pm \sqrt(9)*\sqrt(-1))/2=2i or -i?
Absolutely NOT! Nowhere, is that true.
$\displaystyle \sqrt{1}=1,~\sqrt{1}\ne -1,~\&~-\sqrt{1}=-1$
The fact is: the use of a radical is for real numbers.
Now it is true that for historical reasons we wave our hands and allow $\displaystyle \sqrt{-9}=3i$.
BUT the rules for radicals apply only to real numbers and do not carry over to complex.
Example: $\displaystyle \sqrt{-3}\cdot\sqrt{-3}=\sqrt{3}i\cdot\sqrt{3}i=3i^2=-3$, (hand waving).
Because $\displaystyle \sqrt{-3}$ is not real, we do not apply rules for radicals.
Yes, it exists. The idea with complex functions is to write them in terms of their real and imaginary parts, each of which will be real functions.
Say $\displaystyle \displaystyle \begin{align*} z = x + i\,y = r\,e^{i \left( \theta + 2\pi n \right) } \textrm{ where } r = \sqrt{x^2 + y^2}, \, \theta = \textrm{Arg}\,{(z)} \textrm{ and } n \in \mathbf{Z} \end{align*}$, then
$\displaystyle \displaystyle \begin{align*} f(z) &= \sqrt{z} \\ &= \left[ r\, e^{i \left( \theta + 2\pi n \right) } \right] ^{\frac{1}{2}} \\ &= r^{\frac{1}{2}} \, e^{ i \left( \frac{ \theta }{2} + \pi n \right) } \\ &= \sqrt[4]{x^2 + y^2} \left\{ \cos{ \left[ \frac{\textrm{Arg}\,{(z)}}{2} + \pi n \right] } + i \sin{ \left[ \frac{\textrm{Arg}\,{(z)}}{2} + \pi n \right] } \right\} \\ &= \sqrt[4]{x^2 + y^2}\,\cos{\left[ \frac{\textrm{Arg}\,{(z)}}{2} + \pi n \right] } + i\,\sqrt[4]{x^2 + y^2}\,\sin{ \left[ \frac{\textrm{Arg}\,{(z)}}{2} + \pi n \right] } \end{align*}$