How Can Nested Exponential Integrals Be Solved in Engineering Research?

[navaneethkm]

I am graduate student in engineering. In course of my research I have encountered an integral of this form

$\int_{t'}^{t} e ^{-b t_1} dt_1 \int_{t'}^{t_1 } e ^{b t_2} dt_2 \int_{t'}^{t_2 } e ^{-b t_3} dt_3 \int_{t'}^{t_3} e ^{b t_4} dt_4 ... \int_{t'}^{t_{n-1}} e ^{b t_n} dt_n$

I am trying to find a general form of the result of this integral. Can someone give some pointers on how to solve this integral.Are there problems in physics especially related to translational Brownian motion where one encounters such integrals?

While writing this integral I have assumed t>t1>t2>...>tn>t' .

 

[matteo137]

It's a Dyson series, which in general does not have an analytic solution.
Usually it comes out when in quantum mechanics you want to propagate in time a wave function, according to an Hamiltonian which does not commute with itself at different times.

 

[navaneethkm]

Hi

Thanks. I will read up on this. I was hoping that I would be able to analytically evaluate this integral. Navaneeth