StraRo

Fig.1. \(y=\mathrm{StraRo}_q(x)\) for \(q\!=\!0.5\) , red, for \(q\!=\!1\) , green, and for \(q\!=\!2\) , blue

Fig.2. \(u\!+\!\mathrm i v=\mathrm{StraRo}_{0.5}(x\!+\!\mathrm i y\)

Fig.3. \(u\!+\!\mathrm i v=\mathrm{StraRo}_{1}(x\!+\!\mathrm i y\)

Fig.4. \(u\!+\!\mathrm i v=\mathrm{StraRo}_{2}(x\!+\!\mathrm i y\)

StraRo, or Strange root is function that happened to be useful for evaluation of function ArqNem, that, it its turn, is necessary for evaluation of the Abel function of the Nemtsov Function.

StraRo has singe parameter, denoted gere with letter \(q\).

Description

Let

\(\mathrm{StraR}_q(z)=\sqrt{27 (q\! -\! z)^2 + 4 (1\!+\! 4 q z)^3} \)

In vicinity of the positive part of the real axis,

\(\mathrm{StraRo}_q\) coincides with \(\mathrm{StraR}_q\) .

For \(q\!=\!0.5\), \(q\!=\!1\) and \(q\!=\!2\), for real values of the argument, function \(\mathrm{StraR}_q(x)\) is shown in figure at right with thin curves, \(y=\mathrm{StraR}_q(x)\). The thick curves show \(y=\mathrm{StraRo}_q(x)\), and for positive \(x\), the thin and thick curves overlap.

Function StraR is not so good for construction of function \(\mathrm{ArqNem}_q=\mathrm{Nem}_q^{-1}\), because StraR has cut lines, that limit the range of holomorphizm.

Funiction StraR has cut lines along the negative part of the real axis, and also two segments from zero to the brachpoints \(\mathrm{NemBra}(q)\) and \(\mathrm{NemBra}(q)^*\). There is special article about function NemBra.

For \(q\!=\!0.5\), \(q\!=\!1\) and \(q\!=\!2\), the complex maps of function \(\mathrm{StraR}_q\) are shown in figures 2,3,4:

\(u\!+\!\mathrm i v=\mathrm{StraRo}_q(x\!+\!\mathrm i y)\)

Explicit representation

Let \(Q\) and \(S\) be global variables that store values of \(q\) and \(\mathrm{NemBra}(q)\)

Then, evaluation of \(\mathrm{StraRo}_q(z)\) for complex \(z\) can be performed with the short code below:

z_type F(z_type z){ z_type a,b,r,R ; a=Q-z; b=1. + 4.*Q*z; r = 27.*a*a + 4.*b*b*b ;
DB x,y; x=Re(z); y=Im(z);
if( x<=0 ) return -sqrt(r);
if( y > Im(S) ) return I*sqrt(-r);
if( y < - Im(S) ) return -I*sqrt(-r);
if( x*Im(S)< fabs(y)*Re(S) ) return -sqrt(r);
return sqrt(r);
}

Usage

Function StraRo appears at the inversion of the Nemtsov function \(T\).

Let \(T(z)=\mathrm{Nem}_q(z)=T[z_] = z + z^3 + q z^4\)

Then, the inversion can be performed with Mathematica command

so = Solve[T[A] == z, A]

The Second solution can be extracted with

so2 = ReplaceAll[A, Extract[so, 2]]

The resulting expression has combination (27 q - 27 z)^2 - 4 (-3 - 12 q z)^3, namely, square root of this expression.

Unfortunately, Mathematica does not extract factor 27 from this expression automatically.

FullSimplify[(27 q - 27 z)^2/27 - 4 (-3 - 12 q z)^3/27]

does \(1/27 (729 (q - z)^2 + 108 (1 + 4 q z)^3)\)

instead of expected \(27 (q - z)^2 + 4 (1 + 4 q z)^3\)

However, the simplified expression can be obtained at Mathematica with the manual trick, that bypasses the bug mentioned:

FullSimplify[(27 q - 27 z)^2/27] - 4 FullSimplify[(-3 - 12 q z)^3/27]

that gives expression \(27 (q - z)^2 + 4 (1 + 4 q z)^3\)

One of holomorphic extensions of the square root of this expression is denined as StraRo, and the corresponding value of parameter \(q\) is indicates as subscript. The special, and may be, a little bis strange position of the cut lines of function StraRo is used as a pretext call this function "strange root", and shortening of "strange root" gives the name of the function StraRo.

Keywords

Abel function, ArqNem, AuNem, Book, Mathematica, Nemtsov function, Superfunction, Superfunctions