On 3/2/2014 4:28 AM, G. A. Edgar wrote: >> I have always thought that sqrt(a/b) and sqrt(a)/sqrt(b) are >> exactly the same >> > > Certainly sqrt(a/b) and sqrt(a)/sqrt(b) are both square roots of a/b. > But a complex number has two square roots.
So does a complex number with a zero imaginary part. That is, a real number.
It could happen (as Axel > explains), that the "principal branch" choice for sqrt results in > opposite choices for these two. If a,b are both positive, this does > not happen, and you get the same square root. This assumes, contrary to generally accepted college mathematics "complex analysis" courses, that the sign of the argument of the square root (etc) determines the branch cut. Which it does not. > > Similar things can happen with other powers, with logarithgms, inverse > trig functions, and so on. Unless you choose the arguments nicely > enough.
No, choosing the argument does not choose the branch cut. Unless maybe you are in high school and believe that log(|x|) is a legitimate solution to ignoring the complex plane.
It is possible to choose branch cuts. Students learn to do it. Conformal mapping programs must do it.