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I am trying to solve a problem where I am asked to compute the Taylor expansion series of the function $\sin\left(\frac{1}{1-z}\right)$ around $z=0$. Now, I know that to find the coefficients of the series I can use the Cauchy Integral formula, but here is my doubt: the Taylor series for $\sin(z)$ is valid when $z$ is closed to $0$, and in my case when $z\sim 0$ I have $\sin\left(\frac{1}{1-z}\right) \sim \sin(1)$.

Does everything work the same? Why?

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  • $\begingroup$ The Taylor series for $\sin(z)$ is valid globally, $\sin(z)$ is holomorphic on the whole complex plane. Good luck with the Cauchy integral formula, though, I'm not sure it's really efficient in this case $\endgroup$
    – user436658
    May 14, 2020 at 14:56

3 Answers 3

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You can use Bell polynomials and the Taylor series of $\frac{1}{1-z}$ and $\sin z$, so we have : $$\frac{1}{1-z}=\sum_{n=0}^\infty z^n,\ \sin z=\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$$ Then we have : $$\sin \biggl(\frac{1}{1-z}\biggr)=\sum_{n=0}^\infty\frac{\sum_{k=0}^n \alpha_kB_{n,k}(0!,1!,...,(n-k+1)!)}{n!}z^n$$ Where : $$\alpha_{2n}=0,\\ \alpha_{2n+1}=(-1)^n\frac{1}{(2n+1)!}$$ For more information see the link https://en.wikipedia.org/wiki/Bell_polynomials.

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By the Faa di Bruno formula and some properties for the Bell polynomials of the second kind, we obtain \begin{align*} \biggl[\sin\biggl(\frac{1}{1-z}\biggr)\biggr]^{(n)} &=\sum_{k=0}^{n}\sin^{(k)}\biggl(\frac{1}{1-z}\biggr)B_{n,k}\biggl(\frac{1!}{(1-z)^2},\frac{2!}{(1-z)^3},\dotsc, \frac{(n-k+1)!}{(1-z)^{n-k+2}}\biggr)\\ &=\sum_{k=0}^{n}\sin\biggl(\frac{1}{1-z}+\frac{k\pi}{2}\biggr) \frac{1}{(1-z)^{n+k}} B_{n,k}(1!,2!,\dotsc,(n-k+1)!)\\ &=\sum_{k=0}^{n}\sin\biggl(\frac{1}{1-z}+\frac{k\pi}{2}\biggr) \frac{1}{(1-z)^{n+k}} \binom{n-1}{k-1}\frac{n!}{k!}\\ &\to n!\sum_{k=0}^{n}\sin\biggl(1+\frac{k\pi}{2}\biggr) \binom{n-1}{k-1}\frac{1}{k!} \end{align*} as $z\to0$, where $B_{n,k}(x_1,x_2,\dotsc, x_{n-k+1})$ denotes the Bell polynomials of the second kind. Consequently, we acquire \begin{equation*} \sin\biggl(\frac{1}{1-z}\biggr)=\sum_{n=0}^{\infty}\Biggl[\sum_{k=0}^{n}\sin\biggl(1+\frac{k\pi}{2}\biggr) \binom{n-1}{k-1}\frac{1}{k!}\Biggr]z^n, \quad z\in[-1,1). \end{equation*} Note that the Faa di Bruno formula, the Bell polynomials of the second kind, and their proprties and identies can be found in the following references.

References

  1. Muhammet Cihat Dagli and Feng Qi, Several closed and determinantal forms for convolved Fibonacci numbers, Discrete Mathematics Letters 7 (2021), 14--20; available online at https://doi.org/10.47443/dml.2021.0039.
  2. Bai-Ni Guo and Feng Qi, Viewing some ordinary differential equations from the angle of derivative polynomials, Iranian Journal of Mathematical Sciences and Informatics 16 (2021), no. 1, 77--95; available online at https://doi.org/10.29252/ijmsi.16.1.77.
  3. Feng Qi and Aying Wan, A closed-form expression of a remarkable sequence of polynomials originating from a family of entire functions connecting the Bessel and Lambert functions, Sao Paulo Journal of Mathematical Sciences 15 (2021), in press; available online at https://doi.org/10.1007/s40863-021-00235-2.
  4. Yan Wang, Muhammet Cihat Dagli, Xi-Min Liu, and Feng Qi, Explicit, determinantal, and recurrent formulas of generalized Eulerian polynomials, Axioms 10 (2021), no. 1, Article 37, 9 pages; available online https://doi.org/10.3390/axioms10010037.
  5. Feng Qi, Muhammet Cihat Dagli, and Dongkyu Lim, Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers, Open Mathematics 19 (2021), no. 1, 833--849; available online at https://doi.org/10.1515/math-2021-0079.
  6. Feng Qi, Determinantal expressions and recursive relations of Delannoy polynomials and generalized Fibonacci polynomials, Journal of Nonlinear and Convex Analysis 22 (2021), no. 7, 1225--1239.
  7. Yue-Wu Li, Muhammet Cihat Dagli, and Feng Qi, Two explicit formulas for degenerate Peters numbers and polynomials, Discrete Mathematics Letters 8 (2022), 1--5; available online at https://doi.org/10.47443/dml.2021.0059.
  8. Feng Qi, Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials, Boletim da Sociedade Paranaense de Matematica 39 (2021), no. 4, 73--82; available online at https://doi.org/10.5269/bspm.41758.
  9. Bai-Ni Guo, Dongkyu Lim, and Feng Qi, Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions, AIMS Mathematics 6 (2021), no. 7, 7494--7517; available online at https://doi.org/10.3934/math.2021438.
  10. Feng Qi and Dongkyu Lim, Closed formulas for special Bell polynomials by Stirling numbers and associate Stirling numbers, Publications de l'Institut Mathematique (Beograd) 108 (2020), no. 122, 131--136; available online at https://doi.org/10.2298/PIM2022131Q.
  11. Feng Qi, Pierpaolo Natalini, and Paolo Emilio Ricci, Recurrences of Stirling and Lah numbers via second kind Bell polynomials, Discrete Mathematics Letters 3 (2020), 31--36.
  12. Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contributions to Discrete Mathematics 15 (2020), no. 1, 163--174; available online at https://doi.org/10.11575/cdm.v15i1.68111.
  13. Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Some properties and an application of multivariate exponential polynomials, Mathematical Methods in the Applied Sciences 43 (2020), no. 6, 2967--2983; available online at https://doi.org/10.1002/mma.6095.
  14. Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382.
  15. Feng Qi, Dongkyu Lim, and Bai-Ni Guo, Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations, Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A Matematicas 113 (2019), no. 1, 1--9; available online at https://doi.org/10.1007/s13398-017-0427-2.
  16. Feng Qi, Dongkyu Lim, and Yong-Hong Yao, Notes on two kinds of special values for the Bell polynomials of the second kind, Miskolc Mathematical Notes 20 (2019), no. 1, 465--474; available online at https://doi.org/10.18514/MMN.2019.2635.
  17. Feng Qi, Da-Wei Niu, and Bai-Ni Guo, Some identities for a sequence of unnamed polynomials connected with the Bell polynomials, Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A Matematicas 113 (2019), no. 2, 557--567; available online at https://doi.org/10.1007/s13398-018-0494-z.
  18. Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1.
  19. Feng Qi, Xiao-Ting Shi, Fang-Fang Liu, and Dmitry V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, Journal of Applied Analysis and Computation 7 (2017), no. 3, 857--871; available online at https://doi.org/10.11948/2017054.
  20. Feng Qi and Miao-Miao Zheng, Explicit expressions for a family of the Bell polynomials and applications, Applied Mathematics and Computation 258 (2015), 597--607; available online at https://doi.org/10.1016/j.amc.2015.02.027.
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$\sin\left(\frac{1}{1-z}\right)=\sin\left(1+\frac{z}{1-z}\right)=\sin 1 \cos \left(\frac{z}{1-z}\right) + \cos 1 \sin \left(\frac{z}{1-z}\right),$ so just use Taylor expansions of $\sin$ and $\cos$ around zero for $y=\frac{z}{1-z} \sim 0$.

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