Tanc function: Difference between revisions

Revision as of 20:40, 17 November 2022

Revision as of 20:43, 17 November 2022

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== Properties ==

The first-order derivative of the tanc function is given by

:<math> \frac {1- (\tansec^2(z))^2}{z} - \frac {\tan(z)}{z^2} </math>

The [[Taylor series]] expansion is<math display="block">\operatorname{tanc} z \approx \left(1+ \frac {1}{3} z^2 + \frac {2}{15} z^4 + \frac {17}{315} z^6 + \frac{62}{2835} z^8 + \frac {1382}{155925} z^{10} + \frac{21844}{6081075} z^{12}+ \frac {929569}{638512875} z^{14} + O(z^{16} ) \right)</math>which leads to the series expansion of the integral as<math display="block">\int _0^z \frac {\tan(x) }{x} \, dx = \left(z+ \frac {1}{9} z^3 + \frac {2}{75} z^5 + \frac {17}{2205} z^7 + \frac {62}{25515} z^9+ \frac {1382}{1715175} z^{11}+ \frac {21844}{

79053975} z^{13} + \frac{929569}{9577693125} z^{15}+ O (z^{17}) \right)</math>The [[Padé approximant]] is<math display="block">\operatorname{tanc} \left( z \right) = \left( 1-{\frac {7}{51}}\,{z}^{2} + {\frac {1}{255}}\,{z}^{4}-{\frac {2}{69615}}\,{z}^{6}+{\frac {1}{34459425}}\,{z}^{8} \right) \left( 1-{\frac {8}{17}}\,{z}^{2}+{\frac {7}{255}}\,{z}^{4}-{\frac {4}{9945}}\,{z}^{6}+{\frac {1}{765765}}\,{z}^{8} \right) ^{-1} </math>