Tania function: Difference between revisions
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[[File:TaniaPlot.png|400px|right|thumb|Explicit plot of <math>y=\mathrm{Tania}(x)</math>]] | [[File:TaniaPlot.png|400px|right|thumb|Explicit plot of <math>y=\mathrm{Tania}(x)</math>]] | ||
{{TOC|right}} | {{TOC|right}} | ||
'''Tania function''' is the particular solution ''f = Tania'' of the equation | '''Tania function''' | ||
<ref name="2013or"> | |||
D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326. <br> | |||
http://link.springer.com/article/10.1007/s10043-013-0058-6 <br> | |||
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) | |||
</ref> | |||
is the particular solution ''f = Tania'' of the equation | |||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (1) ~ ~ ~ f'(z)= \frac{f(z)}{1+f(z)} </math> | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (1) ~ ~ ~ f'(z)= \frac{f(z)}{1+f(z)} </math> | ||
satisfying the initial condition | satisfying the initial condition | ||
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==Relation with other special functions== | ==Relation with other special functions== | ||
[[File:ArcTaniaMap70.png|right|350px|thumb| ''f= | [[File:ArcTaniaMap70.png|right|350px|thumb| ''f= ''ArcTania(''x'' + i ''y'') in the ''x,y''-plane with levels ''u'' = '''ℜ'''(''f'') = constant, and ''v'' = '''ℑ'''(''f'') = constant.]] | ||
[[File:TaniaMap100.png|right|350px|thumb| ''f= | [[File:TaniaMap100.png|right|350px|thumb| ''f= ''Tania(''x'' + i ''y'') in the ''x,y''-plane with levels ''u'' = '''ℜ'''(''f'') = constant, and ''v'' = '''ℑ'''(''f'') = constant.]] | ||
The inverse function of | The inverse function of Tania can be called "ArcTania"; ArcTania = Tania<sup>−1</sup>. The ''ArcTania'' is an [[elementary function]]: | ||
: <math>\mathrm{(3)}\ \ | : <math>\mathrm{(3)}\ \ | ||
\mathrm{ArcTania}(z)=z+\ln(z)-1</math> | \mathrm{ArcTania}(z)=z+\ln(z)-1</math> | ||
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~</math> | ~</math> | ||
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. | 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. | ||
After to use, implement and describe the [[Tania function]], it was revealed, that very similar function, [[WrightOmega]] is already described | |||
<ref name="robert"> | |||
http://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf | |||
Robert M.Corless, D.J.Jeffery. On the Wright <math>\omega</math> function. | |||
Univ. of Western Ontario, Canada, (2000) | |||
</ref>: | |||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (6) ~ ~ ~ | |||
\mathrm{WrightOmega}(z)=\mathrm{Tania}(z\!-\!1) | |||
</math> | |||
==History== | ==History== | ||
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Such a model describes the idealized laser amplifier | Such a model describes the idealized laser amplifier | ||
with simple kinetics, and can be applied, for example, to the [[Yb-doped ceramic]]s 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 <math>I_{\rm sat}=100 {\rm W/mm}^2</math>, and the unsaturated gain (that depends on the concentration of dopant and the intensity of pump) may be of order of <math>g_0 \approx 1/\mathrm{mm}</math>.<ref name="unsta"/> This determines the orders of magnitude of values of the physical parameters, while the intensity along the propagation at distance <math>x</math> can be described with intensity <math>I(x)</math> satisfying the equation | with simple kinetics, and can be applied, for example, to the [[Yb-doped ceramic]]s 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 <math>I_{\rm sat}=100 {\rm W/mm}^2</math>, and the unsaturated gain (that depends on the concentration of dopant and the intensity of pump) may be of order of <math>g_0 \approx 1/\mathrm{mm}</math>.<ref name="unsta"/> This determines the orders of magnitude of values of the physical parameters, while the intensity along the propagation at distance <math>x</math> can be described with intensity <math>I(x)</math> satisfying the equation | ||
: <math> \displaystyle ( | : <math> \displaystyle (7) ~ ~ ~ I'(x) = \frac{ g_0 I(x) }{1+I(x)/I_{\rm sat}} </math> | ||
The solution <math>I</math> can be expressed as | The solution <math>I</math> can be expressed as | ||
: <math> \displaystyle ( | : <math> \displaystyle (8) ~ ~ ~ I(x)=I_{\rm sat} \mathrm{Tania}( g_0 x - g_0 x_1)</math> | ||
where <math>x_1</math> is distance at which the intensity saturates the gain, id est, <math>I(x_1)=I_{\rm sat}</math>. | where <math>x_1</math> is distance at which the intensity saturates the gain, id est, <math>I(x_1)=I_{\rm sat}</math>. | ||
The phenomenology of waves with such a saturated gain has been considered in century 20; some results are published in the [[Soviet Journal of Quantum Electronics]].<ref name="kuzne2"/><ref name="kk"/> | The phenomenology of waves with such a saturated gain has been considered in century 20; some results are published in the [[Soviet Journal of Quantum Electronics]].<ref name="kuzne2"/><ref name="kk"/> | ||
The function "Tania" is named after the short name of T.I.Kuznetsova.<ref name="kk"/> | The function "Tania" is named after the short name of T.I. Kuznetsova.<ref name="kk"/> | ||
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).<ref name="tori"/> <!--; the term "Tania function" is suggested after one of the first users of this function.!--> | 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).<ref name="tori"/> <!--; the term "Tania function" is suggested after one of the first users of this function.!--> | ||
Only then the relation with [[WrightOmega]] <ref name="robert"/> | |||
had been revealed. The [[WrightOmega]] function is implemented in many programming languages including [[Mathematica]]. | |||
==Properties of Tania and ArcTania== | ==Properties of Tania and ArcTania== | ||
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The branch points are determined by the singularity of the expression in the right hand side of equation (1), | 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 <math>-1=\exp(\pm \mathrm i \pi)</math>. Using (3), these points can be expressed as follows: | they correspond to values of ArcTania equal to <math>-1=\exp(\pm \mathrm i \pi)</math>. Using (3), these points can be expressed as follows: | ||
: <math>z_{\pm}=\exp(\pm \mathrm i \pi) +\ln(\exp(\pm \mathrm i \pi)) -1=-2\pm \mathrm i \pi</math>. | : <math>\!\!\!\!\!\!\!\!\!(9) ~ ~ ~ z_{\pm}=\exp(\pm \mathrm i \pi) +\ln(\exp(\pm \mathrm i \pi)) -1=-2\pm \mathrm i \pi</math>. | ||
In the left hand side of the figure, between the cut lines lines, the function exponentially approaches zero. | In the left hand side of the figure, between the cut lines lines, the function exponentially approaches zero. | ||
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Tania is real–holomorhic; | Tania is real–holomorhic; | ||
: <math> \mathrm{Tania}(z^*)=\mathrm{Tania}(z)^*</math> | : <math>\!\!\!\!\!\!\!\!\!(10) ~ ~ ~ \mathrm{Tania}(z^*)=\mathrm{Tania}(z)^*</math> | ||
The expansions of Tania in diffrent cases, and, in particular, at the [[branchpoint]]s <math>z_\pm</math> is described below. | The expansions of Tania in diffrent cases, and, in particular, at the [[branchpoint]]s <math>z_\pm</math> is described below. | ||
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At large values of the argument, Tania shows almost linear growth and can be expanded as follows: | At large values of the argument, Tania shows almost linear growth and can be expanded as follows: | ||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle ( | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (11) ~ ~ ~ \mathrm{Tania}(z)=</math> <math>z + 1\!-\!\ln(z) | ||
+\frac{ \ln(z)\!-\!1}{z} </math> <math> | +\frac{ \ln(z)\!-\!1}{z} </math> <math> | ||
+\frac{\ln(z)^2-4\ln(z)+3}{2z^2} </math> <math> | +\frac{\ln(z)^2-4\ln(z)+3}{2z^2} </math> <math> | ||
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+ O\Big( \frac{\ln(z)}{z}\Big)^4 | + O\Big( \frac{\ln(z)}{z}\Big)^4 | ||
</math> | </math> | ||
At <math>|z|\gg 1</math>, the truncated series (7) can be used for the evaluation of < | At <math>|z|\gg 1</math>, the truncated series (7) can be used for the evaluation of <math>\mathrm{Tania}(z)</math>, while the argument is not between the cut lines in figure. | ||
For the evaluation, the asymptotic can be written also as follows: | For the evaluation, the asymptotic can be written also as follows: | ||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! ( | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (12) ~ ~ ~ \mathrm{Tania}(z)=</math> <math> (z\!+\!1)\!-\!\ln(z\!+\!1) | ||
+\frac{ \ln(z\!+\!1)}{z+1}</math> <math> | +\frac{ \ln(z\!+\!1)}{z+1}</math> <math> | ||
+\Big(\frac{ \ln(z\!+\!1)}{z+1}\Big)^{\!2} \Big(\frac{1}{2}-\ln(z\!+\!1)^{-1}\Big) | +\Big(\frac{ \ln(z\!+\!1)}{z+1}\Big)^{\!2} \Big(\frac{1}{2}-\ln(z\!+\!1)^{-1}\Big) | ||
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=== Asymptotic expansion of Tania between the cut lines=== | === Asymptotic expansion of Tania between the cut lines=== | ||
<!-- [[File:TaniaNegMapT.png|350px|right|thumb|Complex map of truncation of series (8)]]!--> | <!-- [[File:TaniaNegMapT.png|350px|right|thumb|Complex map of truncation of series (8)]]!--> | ||
The expansions ( | 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. | ||
For <math>-\Re(z)\!\gg \! 1</math> at <math>|\Im(z)| \!<\! \pi</math> | For <math>-\Re(z)\!\gg \! 1</math> at <math>|\Im(z)| \!<\! \pi</math> | ||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle ( | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (13) ~ ~ ~ \mathrm{Tania}(z) ~\sim~</math> <math>\varepsilon | ||
-\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{7}{2}\varepsilon^4+ O(\varepsilon^5) | -\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{7}{2}\varepsilon^4+ O(\varepsilon^5) | ||
</math> | </math> | ||
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Such expansion can be obtained iterating assignment | Such expansion can be obtained iterating assignment | ||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle ( | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (14) ~ ~ ~ f ~=~ \varepsilon~ \exp(-f)</math> | ||
with the initial approach <math>f=\varepsilon + O(\varepsilon^2)</math>. | with the initial approach <math>f=\varepsilon + O(\varepsilon^2)</math>. | ||
Together with the expansion ( | 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=== | ===Expansion of Tania at singularities=== | ||
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t= \mathrm{i} \sqrt{ \frac{2}{9} (z+2-\pi \mathrm i )~} </math> | t= \mathrm{i} \sqrt{ \frac{2}{9} (z+2-\pi \mathrm i )~} </math> | ||
then, the expansion of Tania can be written as follows: | then, the expansion of Tania can be written as follows: | ||
: <math>\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! ( | : <math>\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (15) ~ ~ ~ \mathrm{Tania}(z)=</math> <math> | ||
-1+3t-3t^2</math> <math> | -1+3t-3t^2</math> <math> | ||
+\frac{3}{4} t^3</math> <math> | +\frac{3}{4} t^3</math> <math> | ||
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The [[Mathematica]] '''Series''' routine easy calculates some tens of the first coefficients of the expansion. | The [[Mathematica]] '''Series''' routine easy calculates some tens of the first coefficients of the expansion. | ||
The series seems to converge while <math>|z\!+2\!-\!\pi \mathrm i |\! <\! 2\pi</math>; in vicinity of another branch point, id est, at | The series seems to converge while <math>|z\!+2\!-\!\pi \mathrm i |\! <\! 2\pi</math>; in vicinity of another branch point, id est, at | ||
<math>z\approx -2\!-\!\mathrm{i}</math>, such an approximation is not valid. | <math>z\approx -2\!-\!\mathrm{i}</math>, such an approximation is not valid. <!-- The figure at right shows [[complex map]] of the truncated series by (10) in the same notations as in the previous maps. The shaded region indicate the domain where the precision of the approximation is less than 3; in the white spot the approximation by the polynomial of 5th order with respect to <math>t</math> gives at least three significant figures. | ||
In vicinity of the origin of coordinates, for example, at <math>-4\!<\!z\!<\!6</math>, the truncated series (with term of order of <math>t^9</math> and higher dropped) provides of order of 2 decimal digits, and the deviation of line <math>v\!=\!0</math> from the abscissa axis is barely seen in the figure.!--> | |||
The figure at right shows [[complex map]] of the truncated series by (10) in the same notations as in the previous maps. | |||
The shaded region indicate the domain where the precision of the approximation is less than 3; in the white spot | |||
the approximation by the polynomial of 5th order with respect to <math>t</math> gives at least three significant figures. | |||
In vicinity of the origin of coordinates, for example, at <math>-4\!<\!z\!<\!6</math>, the truncated series (with term of order of <math>t^9</math> and higher dropped) provides of order of 2 decimal digits, and the deviation of line <math>v\!=\!0</math> from the abscissa axis is barely seen in the figure. | |||
Similar expansion of Tania at point <math>-2-\pi \mathrm i</math> can be obtained by complex conjugation of expressions (9) and (10). | Similar expansion of Tania at point <math>-2-\pi \mathrm i</math> can be obtained by complex conjugation of expressions (9) and (10). | ||
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The [[Taylor expansion]] of Tania at zero | The [[Taylor expansion]] of Tania at zero | ||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle ( | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (16) ~ ~ ~ \mathrm{Tania}(z)=</math> <math>1 </math> <math> | ||
+\frac{z}{2}</math> <math> | +\frac{z}{2}</math> <math> | ||
+\frac{z^2}{16}</math> <math> | +\frac{z^2}{16}</math> <math> | ||
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</math> | </math> | ||
converges at <math>|z|<\sqrt{4+\pi^2}\approx 3.724</math>; this series can be obtained with the [[Mathematica]]'s operator [[InverseSeries]], converting the expansion | converges at <math>|z|<\sqrt{4+\pi^2}\approx 3.724</math>; this series can be obtained with the [[Mathematica]]'s operator [[InverseSeries]], converting the expansion | ||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle ( | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (17) ~ ~ ~ \mathrm{ArcTania}(z)=</math> <math> | ||
2(z\!-\!1)</math> <math> | 2(z\!-\!1)</math> <math> | ||
-\frac{(z\!-\!1)^2}{2}</math> <math> | -\frac{(z\!-\!1)^2}{2}</math> <math> | ||
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+\frac{(z\!-\!1)^6}{6}</math> <math> | +\frac{(z\!-\!1)^6}{6}</math> <math> | ||
+~ {O}(z\!-\!1)^7</math> | +~ {O}(z\!-\!1)^7</math> | ||
However, more terms can be added to the series in ( | However, more terms can be added to the series in (16) and (17). At moderate values of <math>|z|\lesssim 1</math>, | ||
the exansion (11) can be used for the efficient (quick and precise) evaluation of <math>\mathrm{Tania}(z)</math>; | the exansion (11) can be used for the efficient (quick and precise) evaluation of <math>\mathrm{Tania}(z)</math>; | ||
in order to get 14 significant decimal digits, it is sufficient to take 20 terms. | in order to get 14 significant decimal digits, it is sufficient to take 20 terms. | ||
In the similar way, the expansion of <math>\mathrm{Tania}(z)</math> at <math>z=-2+1/\mathrm e\approx -1.632</math> can be written as follows: | In the similar way, the expansion of <math>\mathrm{Tania}(z)</math> at <math>z=-2+1/\mathrm e\approx -1.632</math> can be written as follows: | ||
: <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle ( | : <math>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (18) ~ ~ ~ \mathrm{Tania}(z)=</math> <math> | ||
\frac{1}{\mathrm{e}} </math> <math> | \frac{1}{\mathrm{e}} </math> <math> | ||
+ \frac | + \frac | ||
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The series above can be used for the evaluation of <math>\mathrm{Tania}(z)</math>, dependently on value of <math>z</math>. | The series above can be used for the evaluation of <math>\mathrm{Tania}(z)</math>, dependently on value of <math>z</math>. | ||
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 <math>s_0</math> be initial approximation, obtained with the appropriate formulas from the set above; let | 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 <math>s_0</math> be initial approximation, obtained with the appropriate formulas from the set above; let | ||
: <math> \displaystyle \!\!\!\!\!\!\!\!\!\! ( | : <math> \displaystyle \!\!\!\!\!\!\!\!\!\! (19) ~ ~ ~ s_{n+1}=s_n + \frac{z-\mathrm{ArcTania}(s_n)}{\mathrm{ArcTania}'(s_n)}</math> | ||
where | where | ||
: <math> \displaystyle \!\!\!\!\!\!\!\!\!\! ( | : <math> \displaystyle \!\!\!\!\!\!\!\!\!\! (20) ~ ~ ~\mathrm{ArcTania}'(z)=1+1/z</math> | ||
is derivative of the ArcTania function by equation (3). | 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; | The primary approximations considered in the previous sections allow to get good initial approximation for any point of the complex plane; | ||
then the sequence <math>s</math> converges to <math>\mathrm{Tania}(z)</math>. | then the sequence <math>s</math> converges to <math>\mathrm{Tania}(z)</math>. | ||
The sequence <math>s</math> converges faster than exponentially. Practically, of order of 5 terms in some of the primary approximations are sufficient, and then | The sequence <math>s</math> converges faster than exponentially. Practically, of order of 5 terms in some of the primary approximations are sufficient, and then the third iteration <math>s_3</math> approximates <math>\mathrm{Tania}(z)</math> 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 third iteration <math>s_3</math> approximates <math>\mathrm{Tania}(z)</math> with 15 decimal digits | |||
==Conclusions== | ==Conclusions== | ||
The Tania function came from the [[Laser science]] describing | The Tania function came from the [[Laser science]] <ref name="2013or"> | ||
D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326. <br> | |||
http://link.springer.com/article/10.1007/s10043-013-0058-6 <br> | |||
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) | |||
</ref> | |||
describing | |||
evolution of the light intensity in the [[gain medium]] with simple kinetics. The inverse of Tania function is elementary function. | 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]] <ref name="robert"/> | |||
, 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. | |||
The [[ | |||
==References== | ==References== | ||
{{Reflist|refs= | {{Reflist|refs= | ||
<ref name="unsta"> | <ref name="unsta"> | ||
{{cite journal |url=http:// | {{cite journal |url=http://mizugadro.mydns.jp/PAPERS/2005josab.pdf |author= | ||
D.Kouznetsov, J.-F.Bisson, K.Takaichi, K.Ueda|title=High-power single-mode solid-state laser with a short, wide unstable cavity |journal=J. Opt. Soc. Am. B |volume= 22|issue=8 |pages= pp.1605-1619 |year=2005}} | D.Kouznetsov, J.-F.Bisson, K.Takaichi, K.Ueda|title=High-power single-mode solid-state laser with a short, wide unstable cavity |journal=J. Opt. Soc. Am. B |volume= 22|issue=8 |pages= pp.1605-1619 |year=2005}} | ||
</ref> | </ref> | ||
<ref name="kuzne2"> | <ref name="kuzne2"> | ||
{{cite journal |url=http:// | {{cite journal |url=http://mizugadro.mydns.jp/PAPERS/kk01.pdf |author=T.I.Kuznetsova, D.Yu.Kuznetsov |title=Interaction of a spatially-modulated wave of complex structure and a plane wave in a quantum amplifier |journal=Soviet J. of Quantum Electronics |year=1981 |volume= 11|issue=8 |pages=pp.1090-1094.}}( Russian version: Kvantovaya Elektronika, 1981, v.8, N.8, p.1808-1815.) | ||
</ref> | </ref> | ||
| Line 219: | Line 237: | ||
<ref name="tori"> | <ref name="tori"> | ||
{{cite web |url=http:// | {{cite web |url=http://mizugadro.mydns.jp/t/index.php/Tania_function |title=Tania function |publisher=TORI wiki}} The [http://mizugadro.mydns.jp/t/index.php/Main_Page TORI (Tools for Outstanding Research and Investigation) wiki] is a specialized website to which [http://en.citizendium.org/wiki/User:Dmitrii_Kouznetsov D.Kouznetsov] is a contributor. | ||
</ref>. | </ref>. | ||
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==Note== | ==Note== | ||
Some of the content in this article previously has appeared on the [http:// | Some of the content in this article previously has appeared on the [http://mizugadro.mydns.jp/t/index.php/Main_Page TORI (Tools for Outstanding Research and Investigation) wiki] as [http://mizugadro.mydns.jp/t/index.php/Tania_function Tania function]. In 2013 February-March, [[TORI]] had been attacked and moved from <nowiki>http://tori.ils.uec.ac.jp/TORI</nowiki> to http://mizugadro.mydns.jp/t | ||
[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 06:00, 25 October 2024
Tania function [1] is the particular solution f = Tania of the equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (1) ~ ~ ~ f'(z)= \frac{f(z)}{1+f(z)} }
satisfying the initial condition
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (2) ~ ~ ~ f(0)=1 \ .}
Relation with other special functions
The inverse function of Tania can be called "ArcTania"; ArcTania = Tania−1. The ArcTania is an elementary function:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{(3)}\ \ \mathrm{ArcTania}(z)=z+\ln(z)-1}
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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ArcTania}( \mathrm{Tania}(z))=z}
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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (4) ~ ~ ~ \mathrm{Tania}(z)=\mathrm{LambertW}(\exp(1\!+\!z)) ~} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ ~ |\Im(z)|<\pi}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (5) ~ ~ ~ \mathrm{LambertW}(z)=\mathrm{Tania}\big(\ln(z) -1 \big) ~}
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.
After to use, implement and describe the Tania function, it was revealed, that very similar function, WrightOmega is already described [2]:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (6) ~ ~ ~ \mathrm{WrightOmega}(z)=\mathrm{Tania}(z\!-\!1) }
History
The Tania function is used in the Laser science for the description of intensity of light in the idealized gain medium.
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{\rm sat}=100 {\rm W/mm}^2} , and the unsaturated gain (that depends on the concentration of dopant and the intensity of pump) may be of order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_0 \approx 1/\mathrm{mm}} .[3] This determines the orders of magnitude of values of the physical parameters, while the intensity along the propagation at distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} can be described with intensity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(x)} satisfying the equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle (7) ~ ~ ~ I'(x) = \frac{ g_0 I(x) }{1+I(x)/I_{\rm sat}} }
The solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} can be expressed as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle (8) ~ ~ ~ I(x)=I_{\rm sat} \mathrm{Tania}( g_0 x - g_0 x_1)}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1} is distance at which the intensity saturates the gain, id est, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(x_1)=I_{\rm sat}} .
The phenomenology of waves with such a saturated gain has been considered in century 20; some results are published in the Soviet Journal of Quantum Electronics.[4][5]
The function "Tania" is named after the short name of T.I. Kuznetsova.[5]
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).[6] Only then the relation with WrightOmega [2]
had been revealed. The WrightOmega function is implemented in many programming languages including Mathematica.
Properties of Tania and ArcTania
The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} has two cut lines, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im(z)=\pm \pi~} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\Re(z)\le-2} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1=\exp(\pm \mathrm i \pi)} . Using (3), these points can be expressed as follows:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!(9) ~ ~ ~ z_{\pm}=\exp(\pm \mathrm i \pi) +\ln(\exp(\pm \mathrm i \pi)) -1=-2\pm \mathrm i \pi} .
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;
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!(10) ~ ~ ~ \mathrm{Tania}(z^*)=\mathrm{Tania}(z)^*}
The expansions of Tania in diffrent cases, and, in particular, at the branchpoints Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_\pm} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} should be just Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} . 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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (11) ~ ~ ~ \mathrm{Tania}(z)=} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z + 1\!-\!\ln(z) +\frac{ \ln(z)\!-\!1}{z} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{\ln(z)^2-4\ln(z)+3}{2z^2} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{2 \ln(z)^2 - 15 \ln(z) + 30 \ln(z) -17}{ 6 z^3} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + O\Big( \frac{\ln(z)}{z}\Big)^4 }
At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|\gg 1} , the truncated series (7) can be used for the evaluation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} , while the argument is not between the cut lines in figure. For the evaluation, the asymptotic can be written also as follows:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (12) ~ ~ ~ \mathrm{Tania}(z)=} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z\!+\!1)\!-\!\ln(z\!+\!1) +\frac{ \ln(z\!+\!1)}{z+1}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\Big(\frac{ \ln(z\!+\!1)}{z+1}\Big)^{\!2} \Big(\frac{1}{2}-\ln(z\!+\!1)^{-1}\Big) +\Big(\frac{ \ln(z\!+\!1)}{z+1}\Big)^{\!3} \Big( \frac{1}{3}-\frac{3}{4} \ln(z\!+\!1)^{-1}+ \ln(z\!+\!1)^{-2}\Big)} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\Big(\frac{ \ln(z\!+\!1)}{z+1}\Big)^{\!4} \Big( \frac{1}{4}-\frac{11}{\!5} \ln(z\!+\!1)^{-1}+3\ln(z\!+\!1)^{-2}-\ln(z\!+\!1)^{-3}\Big)} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + O\Big( \frac{\ln(z\!+\!1)}{z+1}\Big)^{\!5} }
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.
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\Re(z)\!\gg \! 1} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Im(z)| \!<\! \pi}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (13) ~ ~ ~ \mathrm{Tania}(z) ~\sim~} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon -\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{7}{2}\varepsilon^4+ O(\varepsilon^5) }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=\exp(1\!+\!z)} .
Such expansion can be obtained iterating assignment
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (14) ~ ~ ~ f ~=~ \varepsilon~ \exp(-f)}
with the initial approach Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=\varepsilon + O(\varepsilon^2)} .
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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} has two cut lines at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(z)\!\le\! -2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im(z)\!=\!\pm \pi} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} 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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{13 z^5}{61440}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{47 z^6}{1474560}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{73 z^7}{41287680}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + ... }
converges at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|<\sqrt{4+\pi^2}\approx 3.724} ; this series can be obtained with the Mathematica's operator InverseSeries, converting the expansion
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (17) ~ ~ ~ \mathrm{ArcTania}(z)=} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(z\!-\!1)} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{(z\!-\!1)^2}{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{(z\!-\!1)^3}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{(z\!-\!1)^4}{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{(z\!-\!1)^5}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\frac{(z\!-\!1)^6}{6}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +~ {O}(z\!-\!1)^7}
However, more terms can be added to the series in (16) and (17). At moderate values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|\lesssim 1} , the exansion (11) can be used for the efficient (quick and precise) evaluation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} ; in order to get 14 significant decimal digits, it is sufficient to take 20 terms.
In the similar way, the expansion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=-2+1/\mathrm e\approx -1.632} can be written as follows:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle (18) ~ ~ ~ \mathrm{Tania}(z)=} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\mathrm{e}} } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + \frac {1/\mathrm{e}} {1+1/\mathrm{e}}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \big(z+2-\frac{1}{\mathrm{e}}\big) } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + \frac{1/\mathrm{e}}{2(1+1/\mathrm{e})^3} \big(z+2-\frac{1}{\mathrm{e}}\big)^2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle + \frac{(1-2/\mathrm{e})/\mathrm{e}}{6(1+1/\mathrm{e})^5} \big(z+2-\frac{1}{\mathrm{e}}\big)^3+...}
The series converges at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \big|z+2-\frac{1}{\mathrm{e}}\big|} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\pi^2+1/\mathrm{e}^2} \approx 3.1445~} .
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tania}(z)} , dependently on value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_0} be initial approximation, obtained with the appropriate formulas from the set above; let
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \!\!\!\!\!\!\!\!\!\! (19) ~ ~ ~ s_{n+1}=s_n + \frac{z-\mathrm{ArcTania}(s_n)}{\mathrm{ArcTania}'(s_n)}}
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \!\!\!\!\!\!\!\!\!\! (20) ~ ~ ~\mathrm{ArcTania}'(z)=1+1/z}
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.)
Conclusions
The Tania function came from the Laser science [1] 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 [2] , 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.
References
- ↑ 1.0 1.1
D.Kouznetsov. Superfunctions for amplifiers. Optical Review, July 2013, Volume 20, Issue 4, pp 321-326.
http://link.springer.com/article/10.1007/s10043-013-0058-6
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) - ↑ 2.0 2.1 2.2 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.)
- ↑ 5.0 5.1 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.
Note
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