Dawson Function

Dawson's function, noted $ F $ or $ D_+ $, also known as Dawson's integral, is a mathematical function defined as $$ F(x)= \exp(-x^2) \int_0^x \exp(y^2)\,\rm{d}y $$

Dawson's function is known to be a particular solution of the differential equation $ y'(x) + 2 x y(x) = 1 $

The graph of the Dawson function is:

The formula for the Dawson function uses an integration, (hence the function is also like a Dawson integral).

It is possible to evaluate the function using an integer series expansion.

Taylor series in $ 0 $: $$ F(x) = \sum_{n=0}^{+\infty} \frac{(-2)^{n}}{1 \cdot 3 \cdot 5 \cdots (2n+1)} \, x^{2n+1} \\ = x - \frac{2}{3} x^3 + \frac{4}{15} x^{5} - \dots + O(x^{2n+1}) $$

Taylor series in $ +\infty $ : $$ F(x) = \frac{1}{2x} + \frac{1}{4x^3} + \frac{3}{8x^5} + \cdots + \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2^{n+1} x^{2n+1}} + O(x^{-2n-1}) $$

The notation $ D_{+}(x) $ for the Dawson function allows to differentiate it, from the symmetric Dawson function $$ D_{-}(x) = \exp(x^2) \int_0^x \exp(-y^2)\,\rm{d}y $$

Dawson's function shares a formula with the erf or erfi error functions $$ F(x)= \frac{\sqrt{\pi}}{2} \exp(-x^2) \operatorname{erfi}(x) = -i \frac{\sqrt{\pi}}{2} \exp(-x^2) \operatorname{erf}(ix) $$

Dawson's function is even, therefore symmetric, $ F(x) = -F(-x) $.

— $ F(0) = 0 $

— $ \lim_{x\to+\infty} F(x) = 0^+ $

— $ \lim_{x\to-\infty} F(x) = 0^- $