تقرير
Unifying trigonometric and hyperbolic function derivatives via negative integer order polylogarithms
| العنوان: | Unifying trigonometric and hyperbolic function derivatives via negative integer order polylogarithms |
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| المؤلفون: | Ducharme, Andrew |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - General Mathematics, 33B10, 33E20, 26A06, 11G55 |
| الوصف: | Special functions like the polygamma, Hurwitz zeta, and Lerch zeta functions have sporadically been connected with the nth derivatives of trigonometric functions. We show the polylogarithm $\text{Li}_s(z)$, a function of complex argument and order $z$ and $s$, encodes the nth derivatives of the cotangent, tangent, cosecant and secant functions, and their hyperbolic equivalents, at negative integer orders $s = -n$. We then show how at the same orders, the polylogarithm represents the nth application of the operator $x \frac{d}{dx}$ on the inverse trigonometric and hyperbolic functions. Finally, we construct a sum relating two polylogarithms of order $-n$ to a linear combination of polylogarithms of orders $s = 0, -1, -2, ..., -n$. Comment: 14 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.19371 |
| رقم الأكسشن: | edsarx.2405.19371 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |