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Γ Function as a Solution of the Equation typeset structure

The typeset structure function can be defined in several ways. One is as a solution of the difference equation

f(z + 1) = z f(z) .(1)

If we assume that the solution is regular at typeset structure then it has simple poles at  typeset structure.The canonical product which vanishes at these points is

h(x) = z Underoverscript[∏, n = 1, arg3] (1 + z/n) e^(-z/n) .

Then

h(z + 1) = (z + 1) Underoverscript[∏, n = 1, arg3] ((1 + z/(n + 1)) e^(-z/n)(1 + 1/n) e^(-1/n)) = h(1)/z h(z),

and

ln h(1) = Underoverscript[∑, n = 1, arg3] (ln(1 + 1/n) - 1/n) = Underscript[lim, n -> ∞] (ln n - 1 - ... - 1/(n - 1)) = -γ .

Thus typeset structure satisfies the difference equation

h(z + 1) = e^(-γ)/z h(z)

and the function

φ(z) = h(z) f(z)

satisfies the difference equation

φ(z + 1) = e^(-γ) φ(z) .

This equation is also satisfied by typeset structure.

If we denote by typeset structure a periodic function with period typeset structure, then

f(z) = (π(z) e^(-γz))/h(z)

represents the general solution of equation (1). For typeset structure identically we get the typeset structure function.

Cite this web-page as:

Štefan Porubský: Γ  Function as a Solution of the Equation typeset structure.

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