Dois teoremas sôbre a função gama

Gomes, Frederico Pimentel

doi:10.1590/S0071-12761956000100006

Resumo

This paper proves the following theorems on the gamma function: Theorem I The integral ∫O∞ t u e-t dt = Γ ( u + 1 ) , where u, real or complex, is such that R (u) > -1, will not change its value if we substitute z = Q (cos φ + i sen φ) for the real variable t, being jconstant and such that - Π/2 < φ < Π/2 , Theorem II The integral ∫-∞∞ w2u + 1 e -w² dw = Γ ( u + 1 ) , where 2u + 1 is supposed to be a non negative even integer, will not change its value if we substitute z = w + fi, f being a real constant, for the real variable w. The proof of both theorems is obtained by means of the well known Cauchy theorem on contour integrals on the complex plane, as suggested by CRAMÉR (1, p. 126) and LEVY (3, p. 178).

Escola Superior de Agricultura "Luiz de Queiroz" Universidade de S. Paulo - Piracicaba

ABSTRACT

This paper proves the following theorems on the gamma function:

Theorem I

The integral

∫_O^∞ t^u e^-t dt = Γ ( u + 1 ) ,

where u, real or complex, is such that R (u) > -1, will not change its value if we substitute

z = Q (cos φ + i sen φ)

for the real variable t, being jconstant and such that

- Π/2 < φ < Π/2 ,

Theorem II

The integral

∫_-∞∞ w^{2u + 1} e -w² dw = Γ ( u + 1 ) ,

where 2u + 1 is supposed to be a non negative even integer, will not change its value if we substitute z = w + fi, f being a real constant, for the real variable w.

The proof of both theorems is obtained by means of the well known Cauchy theorem on contour integrals on the complex plane, as suggested by CRAMÉR (1, p. 126) and LEVY (3, p. 178).

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BIBLIOGRAFIA

(1) CRAMER, Harold - Mathematical Methods of Statistics, la. edição. Princeton University Press, Princeton, 1946.
(2) KENDALL, Maurice C. - The Advanced Theory of Statistics, vol. I, 3a. edição. Charles Griffin, Londres, 1947.
(3) LÉVY, Paul - Calcul des Probabilités, 1a. edição. Gauthier - Villars, Paris, 1925.
(4) PIMENTEL GOMES, Frederico - A Esperança Matemática e Algumas das suas Aplicações à Teoria da Amostragem. An. Esc. Sup. Agr. "Luiz de Queiroz", vol. 8, pp. 638-648, 1951.