How to get smooth square root of complex valued function?

Question

Let $f(x)$ be smooth complex valued function with real argument $x\in[0,1]$. I want to get square root $g(x)=\pm\sqrt{f(x)}$ where the sign is chosen so that g(x) is smooth. How can I do that?

In Mathematica, the branch cut of square root is on negative reals. Thus, for example, I want to flip the sign if $f(x)$ cross negative reals in complex plane when $x$ increases.

Answer 1

The problem can be reduced (I think) to the problem of defining a phase function $\arg(f(x))$ that is continuous on the interval $x \in [0, 1]$. This is tricky, because the path $f(x)$ may "wind" around the origin, leading to different values of the "continuous argument" for the same value of the "conventional argument". A successful "continuous argument" function will need to "keep track of the history" of the function $f(x)$, so that it "knows" whether the phase along the positive real axis should be $0$, $2 \pi$, or something else.

One way to do this is to note that although the conventional Arg function is discontinuous along the negative real axis, its derivative is continuous. Specifically, since $\arg(f(x)) = \Im \ln(f(x))$, we have $$ \frac{d}{dx} \left[ \arg(f(x)) \right] = \Im \left[ \frac{f'(x)}{f(x)} \right]. $$ We can treat this as a differential equation for $\arg(f(x))$; if we integrate it, we'll get a "continuous argument" function. $$ \tilde{\arg}(f(x)) \equiv \arg(f(0)) + \int_0^x \Im \left[ \frac{f'(t)}{f(t)} \right] \, dt. $$ With this in hand, we can then define $$ g(x) = \sqrt{|f(x)|} e^{i \tilde{\arg}(f(x))/2} $$ and this function will be continuous.

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Implementation:

I will test this function on $f(x) = e^{4 \pi i x}$. Difficulties may arise for more complicated functions, particularly those which have roots where $f(x) = 0$. (However, I believe that no smooth $g(x)$ can be defined in such cases anyhow.)

Continuous argument function:

contarg[f_] :=
  Arg[f[0]] + Integrate[Im[f'[t]/f[t]], {t, 0, #}] &;
f[x_] = Exp[4 \[Pi] I x];
Plot[{Arg[f[x]], Evaluate[contarg[f][x]]}, {x, 0, 1}]

Continuous square root:

contsqrt[f_] := Sqrt[Abs[f[#]]] Exp[I contarg[f][#]/2] &
contsqrt[f][x]
Plot[Evaluate[ReIm[contsqrt[f][x]]], {x, 0, 1}]
Plot[Evaluate[ReIm[Sqrt[f[x]]]], {x, 0, 1}, PlotStyle -> Dashed]

(* E^(2 I \[Pi] x) Sqrt[E^(-4 \[Pi] Im[x])] *)

For more complicated functions $f(x)$, Mathematica may not be able to perform the integral required to evaluate contarg[f][x]. In such cases, you may have to resort to using NIntegrate instead.