Tool to calculate the values of the Dawson function (Dawson integral) F(x)=exp(-x^2) ∫_0^x exp(y^2) dy.
Dawson Function - dCode
Tag(s) : Functions
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Dawson Function
Dawson Function Calculator
Answers to Questions (FAQ)
What is the Dawson Function? (Definition)
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: 
How to calculate the Dawson function?
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 $ : $$ F(x) = \frac{1}{2x} + \frac{1}{4x^3} + \frac{3}{8x^5} + \cdots + \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2^{n+1} x^{2n+1}} + O(x^{-2n-1}) $$
Example: $ F(1) \approx 0.53808 $
Why is Dawson's function is noted D+?
The notation $ D_{+}(x) $ for the Dawson function allows to differentiate it, from the symmetric Dawson function $$ D_{-}(x) = \exp(x^2) \int_0^x \exp(-y^2)\,\rm{d}y $$
How to calculate the Dawson function from the error function?
Dawson's function shares a formula with the erf or erfi error functions $$ F(x)= \frac{\sqrt{\pi}}{2} \exp(-x^2) \operatorname{erfi}(x) = -i \frac{\sqrt{\pi}}{2} \exp(-x^2) \operatorname{erf}(ix) $$
What are the properties of Dawson's function?
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^- $
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