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´,

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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**

(8)

Where the **C _{f}** constant is given as

(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)**

(21)

Where is:

(22)

_{f} = 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)**

(31)

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

(32)

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