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Consider the integral

$$\mathcal{I}=\int_0^t\left(\frac{Ae^{-\lambda t}-Ae^{-\lambda s}}{2}\right)^{2m+1}e^{-\epsilon(t-s)}ds,\tag{1}$$

for constants $A,\lambda,\epsilon,t\in\mathbb{R}$ and $m\in\mathbb{Z}^+$.

The intention of evaluating $\mathcal{I}$, is to find

\begin{equation} \begin{split} \mathcal{S}&=\int_0^tJ_1\left(Ae^{-\lambda t}-Ae^{-\lambda s}\right)e^{-\epsilon(t-s)}ds, \\&=\int_0^t\sum_{m=0}^{\infty}\frac{ (-1)^m}{m!(m+1)!}\left(\frac{Ae^{-\lambda t}-Ae^{-\lambda s}}{2}\right)^{2m+1}e^{-\epsilon(t-s)}ds,\\ &=\sum_{m=0}^\infty \frac{ (-1)^m}{m!(m+1)!}\mathcal{I}_m \end{split}\end{equation}

where $J_1$ denotes the first order Bessel function of first kind. Does a closed form exist for $\mathcal{S}$ (or Taylor series)? Could an asymptotic bound simplify things?

According to Mathematica, we have \begin{equation} \mathcal{I}=\frac{(A-Ae^{-\lambda t})^2 (Ae^{-\lambda t}-A)^{2m} 2^{-2m-1}e^{-\epsilon t}\cdot{}_2F_1\left(1,\frac{\epsilon+\lambda}{\lambda};\frac{\epsilon}{\lambda}-2m,e^{-\lambda t}\right)}{A\left(\epsilon-\lambda(2m+1)\right)}. \end{equation}

From this, is it possible to prove that $2\mathcal{S}\sim A\lambda^2\kappa e^{-\lambda t}-A\lambda e^{-\lambda t}$ as $t\rightarrow\infty$, for some $\kappa$?

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  • $\begingroup$ There is no constraint on the sign of $\lambda$? it would be surprising that the asymptotic you suggest (I don't know if it is the right one) works for both positive and negative $\lambda$. $\endgroup$
    – username
    Nov 11, 2021 at 21:59
  • $\begingroup$ The asymptotic is a conjecture based on looking at numerical simulations of the integro-differential equations discussed in mathoverflow.net/questions/407878/…. I have imposed no restrictions for now, but it may very well be incorrect for some certain of $\lambda$. $\endgroup$
    – UNOwen
    Nov 12, 2021 at 7:43

1 Answer 1

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If $\lambda>0$ and $\lambda>\epsilon$ one has $$\lim_{t\rightarrow\infty}e^{\epsilon t}\mathcal{S}=\int_0^\infty J_1\left(-Ae^{-\lambda s}\right)e^{\epsilon s}ds$$ $$\qquad=\frac{A/2}{\epsilon-\lambda} \, _1F_2\left(\frac{1}{2}-\frac{\epsilon}{2 \lambda};2,\frac{3}{2}-\frac{\epsilon}{2 \lambda};-A^2/4\right).$$ The asymptotics is different for other ranges of $\lambda,\epsilon$, for example, for $\lambda,\epsilon<0$ one has ${\cal S}\rightarrow -e^{-\epsilon t}J_1(Ae^{-\lambda t})/\epsilon$.

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