How to expand \begin{align*} y^{n} \mathrm{e}^{\mathrm{i}(y-c)^{2}} \end{align*} in infinite series? Please help me for this.The series exansion of \begin{align*} y^{n} \mathrm{e}^{\mathrm{i}y^{2}} \end{align*} is possible but I can not find for the above function.
Series expansion of a function
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$\begingroup$
sequences-and-series
taylor-expansion
1 Answers
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Note that $\exp(i(y-c)^2)=\exp(iy^2-2icy)\cdot\exp(ic^2)$ and $$\exp(uy^2+vy)=\sum_{k=0}^\infty\frac1{k!}(uy^2+vy)^k=\sum_{k=0}^\infty\frac1{k!}y^k(uy+v)^k= \sum_{k=0}^\infty a_ky^k$$ where $$a_k=\sum_{r=0}^{\lfloor k/2\rfloor} \frac{{k-r\choose r}u^rv^{k-2r}}{(k-r)!}$$ (because ${k-r\choose r}u^rv^{k-2r}$ is the summand of $y^r$ in $(uy+v)^{k-r}$).