Is there a way of approximating e^-x for large x

[Chain]

I'm trying to evaluate an integral with e^-x where x is huge in the domain of the integral so I can't evaluate it numerically without making an approximation.

 

[mfb]

Can you add more context? There might be a clever way to approximate e^(-x), but if that value is not added to something, e^(-x)=e^(-x-x₀)*e^(x₀) where the second factor is independent of x and the first factor can be chosen to be about e^0.

 

[Chain]

The integral is 4*(r^2)*exp(-2*r/a)/a^3 integrated between 1 and ∞ (The probability an electron in the ground state of hydrogen is more than 1 metre away from the nucleus) a=0.529*10^-10

 

[Hurkyl]

Note that your integrand is one you can anti-differentiate without much trouble, so you can get the exact answer. (e.g. integration by parts. Or computer algebra package)

 

[mfb]

If you just want an estimate: The exponential will drop very quick, so regions with r>1m+eps are irrelevant and r^2 is nearly constant and =1m^2. Therefore, the integral is simply 4m^2/a^3 * exp(-2r/a) which can be evaluated as 2m^2/a^2 * exp(-2m/a) ≈ 8*10^20 * exp(-4*10^10) ≈ 10^(-10^10) where the last approximation is very rough.

 

[mathman]

e^-x = 10^{-0.43429448190325x}. The constant is log₁₀e.

Using the above for large x, you can separate the integer and fractional parts of the exponent. I assume you know how to proceed from here.

 

[lurflurf]

Are you talking about Gauss–Laguerre quadrature?

\int_0^\infty f(x) e^{-x} \mathop{\text{dx}}\sim \sum_{i=1}^n w_i f(x_i)

where xi are zeros of a Laguerre polynomial and

w_i=\frac{x_i}{(n+1)^2[L_{n+1}(x_i)]^2}

 

[Chain]

Fair enough, yeah I realized after posting this the integral could be solved analytically >__< and I got a valule of something like 10^(-10^10) but thanks for the responses :)