Abstract

In the article will be shown that factorial can be computed by the following formula:

(30)

Gamma constant can be defined by the following formula:

(27)

Gamma constant has following numerical value γ = 0.57721566490153286060651209008.


Author of the following article is Andrija S. Radovic, Andrija Radovic, all rights reserved. The parts or whole article can not be published without author's prior agreement and without the author's name.

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A REPRESENTATION OF FACTORIAL FUNCTION, THE NATURE OF γ CONSTANT AND A WAY FOR SOLVING OF FUNCTIONAL EQUATION F(x) = x F(x - 1)

The Γ function is given as recursive relation by the definition:

(1)

Derivative of the recursive relation is given as:

(2)

(3)

(4)

(5)

(6)

(7)

If the recursive process is continued to ∂((n-n)!)/∂n it will be obtained:

(8)


Where the Cf constant is given as . Now it is possible to be obtained a series of equations that every single one is derived by a previous:

(9)

(10)

(11)

This sum can be summarized regarding the next formula that defines the sum of geometric series:

(12)

Now, the next equations is given that may be applied for the summarization of the series:

(13)

(14)

(15)

The final form of the formula for the series summarization is:

(16)

Now It can be derived the formula of the factorial function:

(17)

(18)

Integrals may swap the places and thus the next formula is obtained:

(19)

Anent,

(20)

Now, Ψ(x) can be defined on the next way; while the next equation is valid:

(21)

Where is:

(22)

Cf = γ

(23)

Let we observe the next expression:

(24)

With respect to the upper expression for factorial, it can be exanimated the nature of the γ constant. See the formula:

(25)

Let is n = 1:

(26)

(27)

(28)

(29)

The next numeric formula for factorial computing is derived by the equations (19) and (26):

(30)

The next formula for computing of the Ψ(x) is obtained with respect to the formulas (21) and (30):

(31)

By the formula (30) it can be computed the numerical solution of SQR(-1)!:

(32)

i! = 0.49801566811835604271 - 0.15494982830181068513 i


Links to relevant pages containing above formulas:

http://functions.wolfram.com/Constants/EulerGamma/09/0010/

http://functions.wolfram.com/GammaBetaErf/Gamma/07/01/01/0006/

http://functions.wolfram.com/GammaBetaErf/PolyGamma/07/01/01/0005/

http://functions.wolfram.com/GammaBetaErf/Gamma/07/01/01/0005/


Author:
Dipl. Ing. Andrija S. Radović
Tel: +381(64)1404075

E-mails:
andrijar@andrijar.com
or
andrija_radovic@andrijar.com