Final answer:
The error function, erf(x), can be defined as an integral. To obtain an asymptotic expansion for erf(x) as x ≥ 0, we can use the power series expansion of the exponential function. As x approaches infinity, we can use Laplace's method to find the asymptotic expansion.
Step-by-step explanation:
The error function, erf(x), can be defined as an integral:
erf(x) = (2/√π) ÞE;x0 e-t2 dt
To obtain an asymptotic expansion for erf(x) as x ≥ 0, we can use the power series expansion of the exponential function:
ex = 1 + x + (x2/2!) + (x3/3!) + ...
By substituting this series into the integral for erf(x) and simplifying, we can obtain the asymptotic expansion.
As x approaches infinity, we can use a different expansion technique called Laplace's method. This involves finding the maximum of the exponent in the integrand and then using a Taylor series expansion around that maximum.
By applying these methods, we can obtain the asymptotic expansions for erf(x) as x ≥ 0 and as x approaches infinity.