# Integral of $J_1\left(Ae^{-\lambda t}-Ae^{-\lambda s}\right)e^{-\epsilon(t-s)}$ with respect to $s$?

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

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

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 $$$$\mathcal{I}=\frac{(A-Ae^{-\lambda t})^2 (Ae^{-\lambda t}-A)^{2m} 2^{-2m-1}e^{-\epsilon t}\cdot{}_2F_1\left(1,\frac{\epsilon+\lambda}{\lambda};\frac{\epsilon}{\lambda}-2m,e^{-\lambda t}\right)}{A\left(\epsilon-\lambda(2m+1)\right)}.$$$$

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 $$\kappa$$?

• 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 $\lambda$. Nov 11, 2021 at 21:59
• 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 $\lambda$. Nov 12, 2021 at 7:43

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 $${\cal S}\rightarrow -e^{-\epsilon t}J_1(Ae^{-\lambda t})/\epsilon$$.