What is the Integral Transport Function?
The integral transport function — usually written En(x) — is a family of special functions defined by an integral of the form ∫1∞ e−xt t−n dt. It appears throughout radiative transfer, neutron transport theory, kinetic theory, and asymptotic analysis. Despite the name, it is essentially a scaled exponential integral and shares many of its properties.
Mathematical definition
For an integer order n ≥ 1 and a real argument x ≥ 0:
Equivalently, by the substitution u = x t:
where Γ(s, x) is the upper incomplete gamma function. This identity is the basis of the calculator's algorithm: we evaluate the upper incomplete gamma function numerically and apply the power scaling, rather than integrating directly — this is far more stable for large x.
Relevant parameters
- Order n — a positive integer. It controls the power-law decay of the integrand. Larger n makes the integrand decay faster near t = 1, but also introduces stronger cancellation in the recurrence relations.
- Argument x — a non-negative real number. It sets the exponential decay rate. For x = 0 the integral reduces to 1/(n−1); for large x the function decays like e−x/x.
How the function is evaluated
This calculator uses a robust numerical scheme:
- Boundary cases — x = 0 returns 1/(n−1) exactly. x larger than ~100 returns 0 (underflow) with a note.
- Recurrence for small n — for n = 1 we use the exponential integral E1(x) computed via a minimax rational approximation (Abramowitz & Stegun 5.1.53–5.1.56).
- Forward recurrence — for n ≥ 2 we use n En+1(x) = e−x − x En(x), starting from E1(x). This is stable for small n (≤ 20) and moderate x.
- High-precision fallback — for n > 20 or when recurrence loses accuracy, we evaluate the integral representation xn−1 Γ(1−n, x) using a continued fraction expansion of the incomplete gamma function (Lentz's algorithm).
The result is accurate to at least 10 significant digits for the typical range 1 ≤ n ≤ 20, 0 ≤ x ≤ 100, and degrades gracefully outside it with an explicit warning.
Applications
- Radiative transfer — the En(τ) functions (closely related) describe the exponential integral moments used in the Schwarzschild–Milne equations.
- Neutron transport — the transport function appears in the solution of the neutron transport equation for slab geometry.
- Plasma physics — the exponential integral family describes collisional damping and runaway electron dynamics.
- Heat conduction — some integral solutions of the heat equation reduce to En forms.
- Asymptotic analysis — the function is a standard test case for uniform asymptotic expansions.
Numerical considerations
Direct numerical integration of ∫1∞ e−xt t−n dt is possible but inefficient for large x, because the integrand becomes extremely peaked near t = 1. The recurrence method used here avoids that issue by reducing the problem to the well-studied exponential integral. However, forward recurrence is known to be unstable for large n (roughly n > 20), which is why we switch to a continued-fraction evaluation of Γ(1−n, x). The calculator also checks for overflow/underflow and returns an explicit note rather than a silently wrong value.
Worked example
Let us compute E2(1).
Using the recurrence E2(x) = (e−x − x E1(x)) / 1 with x = 1 and the known value E1(1) ≈ 0.219383934:
The calculator returns 0.1484955068… (10+ digits).
FAQ
What is the difference between En(x) and the Chandrasekhar En(τ)?
The Chandrasekhar moment is often defined as ∫01 μn−2 e−τ/μ dμ, which equals En(τ) times 1/(n−1)! for integer n. This calculator uses the exponential-integral convention, which is the most common in mathematical tables (Abramowitz & Stegun).
Can I compute non-integer orders?
The integral definition extends to real n > 1, but this calculator is validated and optimized for integer n from 1 to 20. For non-integer orders, use the incomplete gamma function directly.
Why does x = 0 give a finite value?
At x = 0 the exponential factor vanishes and the integral reduces to ∫1∞ t−n dt = 1/(n−1) for n > 1. For n = 1 it diverges, which is correctly flagged as an error.
What if x is very large?
For x > 100 the function underflows double precision (≈ e−100 ~ 10−44), so we return 0 and show a note. You can still compute ratios or use asymptotic expansions manually.