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Dawson Function

Tool to calculate the values of the Dawson function (Dawson integral) F(x)=exp(-x^2) ∫_0^x exp(y^2) dy.

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Dawson Function -

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# Dawson Function

## Dawson Function Calculator

### 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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Dawson Function on dCode.fr [online website], retrieved on 2023-09-27, https://www.dcode.fr/dawson-integral-function

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