^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{3}

In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chèbyshev polynomials of the first kind and the error function. Also we provide its series representation using Padé approximant. We show a convincing numerical evidence about an accuracy of

Integrals of the error function, see (^{1}

Now, if we really need a simple expression for

The function is very nice. It goes to its limit at

The plot of

Therefore (depending on the accuracy we need) we can easily take

Mathematica gives the following for the first

Now, what can we do for small

The function is so nice; we can just use the Taylor expansion around

The plot for

Approximation of

Now let us find a better approximation by computing the higher derivatives.

We use Mathematica as a shortcut, but it is easy to do it by hand, if we remember that

so our next approximation is as follows.

The plot with both approximations (orange, green) and the function itself (blue) is given in Figure

Plot of I(a) with both approximations.

but this is an ordinary differential equation, which can be solved numerically.

As an illustration, here is a simple explicit Euler scheme for the step size

We can use an initial value

For

A simple explicit Euler scheme for the step size

For

A simple explicit Euler scheme for the step size

This way can serve as a good alternative to numerical integration [

The well-known formula which expresses the relationship between error function and cumulative density function, see (^{2}

And it is easy to check that it always holds for every real number by the following short proof.

By definition, the error function

Writing

Those integrals on the right hand size are both values of the CDF of the standard normal distribution:

Specifically,

It is a theorem of Liouville [

can be expressed in terms of elementary functions if and only if there exists some rational function

We may give here a possible approach formula for T(

Error approximation for T(

We may try to find a series expansion in powers of

Convergence of

Series expansion of

Suppose that we have the Taylor expansions:

Then we have the standard result:

where

Taylor expansion of

Since

Now by composition of (

n-th derivative of new special function

Series expansion of new special function

We have the power series of

We fully agree that

Now we are ready to approximate

Short table for approximation comparison.

n | approximation | exact |
---|---|---|

1 | 0.591506 | 0.596751 |

2 | 0.279674 | 0.283168 |

3 | 0.151067 | 0.153256 |

4 | 0.0870954 | 0.0884650 |

5 | 0.0522216 | 0.0530855 |

6 | 0.0321485 | 0.0326982 |

7 | 0.0201718 | 0.0205243 |

8 | 0.0128409 | 0.0130686 |

9 | 0.00826756 | 0.00841548 |

10 | 0.00537202 | 0.00546863 |

Recall

Reasonable approximation using bounds.

k | approximation | exact |
---|---|---|

1 | 0.22870436048 | 0.22959937502 |

2 | 0.08960938943 | 0.08997882179 |

3 | 0.04400808083 | 0.04418398568 |

4 | 0.02389675159 | 0.02398719298 |

5 | 0.01374034121 | 0.01378897319 |

6 | 0.00819869354 | 0.00822557475 |

7 | 0.00502074798 | 0.00503586007 |

8 | 0.00313428854 | 0.00314286515 |

9 | 0.00198581489 | 0.00199069974 |

10 | 0.00127304507 | 0.00127582211 |

The function

We may approximate the function

Let

Define

We look for

Adjusting the error function by means of the new special function.

In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used for describing particle speeds in idealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment. The term “particle” in this context refers to gaseous particles (atoms or molecules), and the system of particles is assumed to have reached thermodynamic equilibrium. The energies of such particles follow what is known as Maxwell–Boltzmann statistics, and the statistical distribution of speeds is derived by equating particle energies with kinetic energy. Mathematically, the Maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom (the components of the velocity vector in Euclidean space), with a scale parameter measuring speeds in units proportional to the square root of

Maxwell–Boltzmann distribution.

Approximation of the CDF for the Maxwell–Boltzmann distribution for

Finally this approximation by new special function showed that it may also be applied in thermodynamics to evaluate the average energy per particle in the circumstance where there is no energy-dependent density of states to skew the distribution, and the representation of probability for a given energy must be normalized to a probability of 1 which holds using our new special function with two parameters as shown in (

We have studied A new probability distribution defined on

The data supporting this new special function is the Table of integral of error functions which is cited as one of the important reference in this paper and that function dosn't montioned in any standard references related to the integral of error function.

Zeraoulia Rafik present address is Department of Mathematics, High school-Timgad, Batna, Algeria.

There are no conflicts of interest regarding the publication of this paper.

Zeraoulia Rafik, Alvaro H. Salas, and David L. Ocampo are equally contributing authors.

Alvaro H. Salas, Universidad Nacional de Colombia, Colombia, has received the Grant/Award Number 283,

In mathematics, the error function (also called the Gauss error function) is a special function (nonelementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as

Cumulative distribution function for the normal distribution. In probability theory and statistics, the cumulative distribution function (CDF, also cumulative density function) of a real-valued random variable