- 1 Relation with other special functions
- 2 History
- 3 Properties of Tania and ArcTania
- 4 Evaluation of Tania function
- 5 Conclusions
- 6 References
- 7 Note
Tania function  is the particular solution f = Tania of the equation
satisfying the initial condition
Relation with other special functions
The inverse function of Tania can be called "ArcTania"; ArcTania = Tania−1. The ArcTania is an elementary function:
The complex maps of f = ArcTania(z) and f = Tania(z) are shown in figures at right in the plane x = ℜ(z), y = ℑ(z) with lines u = ℜ(f) = constant, and lines v = ℑ(f) = constant. Symbol ℜ denotes "real part" and symbol ℑ denotes "imaginary part".
At least for |ℑ(z)| < π, the relation below holds
In certain range of values of the argument, the Tania function can be expressed through the Lambert W-function (known also as the ProductLog function) as follows:
At least in vicinity of the real axis, the Tania function can be expressed through the Lambert W-function; while the principal branch of the Lambert W-function can be expressed through the Tania function for all complex values of the argument.
The Tania function appears as the intensity of a wave in the uniformly pumped gain medium, measured in unit of the saturation intensity. Then, its argument has sense of length of propagation, measured in units of the inverse of unsaturated gain. Such a model describes the idealized laser amplifier with simple kinetics, and can be applied, for example, to the Yb-doped ceramics pumped in vicinity of wavelength of order of 940nm; the maximum gain corresponds to wavelength in vicinity of 1030 nm. For this wavelength, the saturation intensity may have value of order of , and the unsaturated gain (that depends on the concentration of dopant and the intensity of pump) may be of order of . This determines the orders of magnitude of values of the physical parameters, while the intensity along the propagation at distance can be described with intensity satisfying the equation
The solution can be expressed as
where is distance at which the intensity saturates the gain, id est, .
The function "Tania" is named after the short name of T.I. Kuznetsova.
For the applications, the simple and robust implementation of the Tania function is an important tools of the scientific research. For this reason, the description of the Tania function is loaded at TORI (Tools for Outstanding Research and Investigations). Only then the relation with WrightOmega  had been revealed. The WrightOmega function is implemented in many programming languages including Mathematica.
Properties of Tania and ArcTania
The has two cut lines, , . In the figure, these lines are marked with dashed half-lines. The branch points are determined by the singularity of the expression in the right hand side of equation (1), they correspond to values of ArcTania equal to . Using (3), these points can be expressed as follows:
In the left hand side of the figure, between the cut lines lines, the function exponentially approaches zero. In other directions, the modulus of the function is smoothly growing; the growth is similar to that of the linear function.
Tania is real–holomorhic;
The expansions of Tania in diffrent cases, and, in particular, at the branchpoints is described below.
Expansion of Tania at infinity
The complex maps of Tania and ArcTania above show that asymptotically each of these functions becomes similar to its argument. Being far from the branch points (and outside the cut lines), the grid of the lines of constant real and constant imaginary part looks similar to that of the identity function. This indicates that the leading term of the expansion at infinity of should be just . Such an expansion is suggested in this section.
At large values of the argument, Tania shows almost linear growth and can be expanded as follows:
At , the truncated series (7) can be used for the evaluation of , while the argument is not between the cut lines in figure. For the evaluation, the asymptotic can be written also as follows:
Asymptotic expansion of Tania between the cut lines
The expansions (11) and (12) are not valid between the cut lines. For this range, for large values of the argument, the special expansion is suggested in this section.
Such expansion can be obtained iterating assignment
with the initial approach .
Together with the expansion (11), the expansion (13) covers the most of the complex plane, leaving only a finite-size domain that includes the branch points and the origin of the coordinates. The expansion at these points are considered below.
Expansion of Tania at singularities
has two cut lines at , . The expansions above do not reproduce the behavior of Tania in vicinity of the branch points. In this section, the expansion at these branch points is suggested.
For the expansion of at the upper branch point , the convenient small parameter is
then, the expansion of Tania can be written as follows:
The Mathematica Series routine easy calculates some tens of the first coefficients of the expansion. The series seems to converge while ; in vicinity of another branch point, id est, at , such an approximation is not valid.
Similar expansion of Tania at point can be obtained by complex conjugation of expressions (9) and (10).
Taylor expansions of Tania
The expansions above, in principle, allow to cover all the complex plane with the approximations of the Tania function. But the expansion at the branch points may be not convenient while dealing with Tania of a real argument. For this case, the Taylor expansions of Tania at the real axis are suggested in this section.
The Taylor expansion of Tania at zero
However, more terms can be added to the series in (16) and (17). At moderate values of , the exansion (11) can be used for the efficient (quick and precise) evaluation of ; in order to get 14 significant decimal digits, it is sufficient to take 20 terms.
In the similar way, the expansion of at can be written as follows:
The series converges at .
The truncated expansions above cover all the complex plane with at least 3–digit approximations.
Evaluation of Tania function
The series above can be used for the evaluation of , dependently on value of . The increase of the precision of the evaluation is easier to achieve not with the increase of number of terms in the expansion, but applying the Newton method. Let be initial approximation, obtained with the appropriate formulas from the set above; let
is derivative of the ArcTania function by equation (3). The primary approximations considered in the previous sections allow to get good initial approximation for any point of the complex plane; then the sequence converges to .
The sequence converges faster than exponentially. Practically, of order of 5 terms in some of the primary approximations are sufficient, and then the third iteration approximates with 15 decimal digits. Few tens operations are sufficient to get the maximal precision available with the complex(double) variables; the graphics can be plotted in the real time with the C++ generators supplied. (Click the figure to see the implementation.)
The Tania function came from the Laser science  describing evolution of the light intensity in the gain medium with simple kinetics. The inverse of Tania function is elementary function. The Tania function can be expressed through the function WrightOmega , but the C++ complex(double) implementation Tania.cin is already prepared; it is used for generators of the figures. The Lambert W function (known also as ProductLog function) can be expressed through the Tania function as well as through the WrightOmega function.
D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.
http://mizugadro.mydns.jp/PAPERS/2013or1.pdf single column version with links for online reading (12 pages) http://mizugadro.mydns.jp/PAPERS/2013or2.pdf two column version for printing (6 pages)
- http://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf Robert M.Corless, D.J.Jeffery. On the Wright function. Univ. of Western Ontario, Canada, (2000)
- D.Kouznetsov, J.-F.Bisson, K.Takaichi, K.Ueda (2005). "High-power single-mode solid-state laser with a short, wide unstable cavity". J. Opt. Soc. Am. B 22 (8): pp.1605-1619.
- T.I.Kuznetsova, D.Yu.Kuznetsov (1981). "Interaction of a spatially-modulated wave of complex structure and a plane wave in a quantum amplifier". Soviet J. of Quantum Electronics 11 (8): pp.1090-1094.. ( Russian version: Kvantovaya Elektronika, 1981, v.8, N.8, p.1808-1815.)
- T.I.Kuznetsova, D.Yu.Kuznetsov (1984). "Interaction between counterrunning spatially modulated beams in a non-linear medium". Soviet J. of Quantum Electronics 14 (11): pp.1471-1474. (Russian version: On the interaction between counterpropagating spatially modulated beams in a non-linear medium. -- Kvantovaya Elektronika, 1984, v.11, N.11, p.2210-2215))
- Tania function. TORI wiki. The TORI (Tools for Outstanding Research and Investigation) wiki is a specialized website to which D.Kouznetsov is a contributor.
Some of the content in this article previously has appeared on the TORI (Tools for Outstanding Research and Investigation) wiki as Tania function. In 2013 February-March, TORI had been attacked and moved from http://tori.ils.uec.ac.jp/TORI to http://mizugadro.mydns.jp/t