With Matlab, I want to verify numerically whether the following inequality holds
$$\exp(-27^2)\text{erfi}(27)<\frac{21}{1000}.$$
However, I obtain "Inf" with Matlab. The reason is due to overflow, which lead to results too large to represent as conventional floating-point values.
My question is: Is there some trick to compute the LHS with Matlab?
Thanks for any help!
So we want to prove that
$$ \frac{2}{\sqrt{\pi}}\int_{0}^{27} e^{x^2-27^2}\,dx < \frac{21}{1000}\tag{1} $$
but plenty of continued fraction representations for the Dawson function are already known.
For instance, for any $x\geq 2$
$$ \text{erfi}(x)e^{-x^2} \leq \frac{2x}{\sqrt{\pi}}\cdot\frac{1}{1+2x^2-\frac{4x^2}{2x^2-1}}\tag{2} $$
holds, and the evaluation at $x=27$ leads to
$$ \frac{2}{\sqrt{\pi}}\int_{0}^{27} e^{x^2-27^2}\,dx=\frac{2}{\sqrt{\pi}}\int_{0}^{27}e^{x^2-54x}\,dx \leq 0.02092.\tag{3} $$
The middle representation is best suited for numerical purposes: $e^{x^2-54x}$ takes values $\geq 10^{-20}$ only over $[0,1]$, and $\int_{0}^{\color{red}{1}}e^{x^2-54x}\,dx $ is straightforward to approximate through Maclaurin series, convexity inequalities like $e^{x^2}\leq 1+x^2+(e-2)x^4$, Cauchy-Schwarz etc.
