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Gauss's Definition of the Gamma Function

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Gauss's Definition of the Gamma Function ¢¡£¡£¤¦¥¨§£© ¢ ¤¦ £§£¤¦§¤¦§£ £¤¦££ ¤¦§£¤¦ §£¡£©£© "!$# %'&( )"*,+ - # %'./%0*,1 2 3£%0*,# 45%7698 8 :; -=< 2@?A./"4 > B7¤¦C£©£©DE© £§ FG¡£ HG¡IJFHGK£§L£¤¦¤FGC£¡£ MHG¡ U VEW S X NPORQTS The present paper gives a historical account on extending the factorial function to complex numbers by Gauss. Y7Z0[0\0]_^a`0b0cPd0]EZ0e \0fge fP^T\g[0e ZP^T\0]Ehi\PjacP\0kJl0m `0bgf0d0kJn0`0]EZ Complex numbers (more exactly, square roots of negative numbers) have been used since the mid of the 16th century in mathematics. These numbers appeared when Italian mathematicians tried to solve polynomial equations of higher degrees. The use of complex numbers sometimes led to certain obscuri- ties. A change of the perception of the complex numbers came around the turn of the 18th and 19th century when Karl Friedrich Gauss (1777-1855) published several papers where he used an idea of complex numbers. Further progress came with representation of complex numbers by points or vectors in the plane. This idea occurred for the first time in the works of Caspar Wessel (1745-1818) and Jean Robert Argand (1768-1822); Argand introduced the term textitle module for an absolute value of the complex numbers. Both Wessel's and Argand's articles were written probably independently and these works have never received general awareness. The geometric interpretation of complex numbers was completely accepted when Gauss wrote his treatise Theo- ria residuorum biquadraticorum (Theory of the biquadratic residues) (1831), see [1]. In 1837 William Rowan Hamilton (1805-1865) introduced complex num- bers as ordered pairs of real numbers. In 1847 Louis Augustin Cauchy (1789- 1857) presented an algebraic definition of complex numbers. Y7o"pqd0Z0Zsr Zm `P^ ^T`0]=^T\it0`0Z0Z0`0m On 21st November 1811 Karl Friedrich Gauss wrote a letter to Friedrich Wilhelm Bessel (1784-1846) about his development on general factorials, see [3]. Gauss wrote the following text in that letter (authors translation). AMS (2000) Subject Classification: 01A55. uwvxwy{zP| }_~0"@ Thus the product 1 · 2 · 3 · : : : · x = x Y is the function that, in my opinion, must be introduced into Calculus, (. ). But if one wants to avoid countless Kramp's paradigms and para- doxes and contradictions, 1·2·3·: : : ·x must not be used as the definition of x, because such a definition has a precise meaning only when x is anQinteger; rather, one must start with a more general definition, which is applicable even to imaginary values of x, of which that one is a special case. I have chosen the following one x 1 · 2 · 3 · : : : · k · k x = ; x x x · : : : · x k Y ( + 1)( + 2)( + 3) ( + ) where k tends to infinity. Let us remark that Christian Kramp (1760-1826) was one of the mathe- maticians who sought a general rule for non-integer values of factorials. He introduced so-called “numerical factorial” by the form b a c = a(a + c)(a + 2c) · : : : · (a + (b − 1)c) (1) in the book [4]. The product on the right-hand side of (1) was studied in the first half of the 19th century under the name “analytic factorial”. In 1856 Karl Weierstrass (1815-1897) finished these activities, when he demonstrated nonsense resulting from that definition, see [8]. Moreover, Kramp was the first mathematician who used the notation n! for n-factorials in the book [5]. Gauss acquired his knowledge about the function x during an investiga- tion of properties of the hypergeometric series. The generalizedQ hypergeometric function pFq(α1; : : : ; αp; β1; : : : ; βp; x) 1 k is defined by the sum of a hypergeometric series, i.e., series k=0 akx for a ak+1 which 0 = 1 and the ratio of consecutive terms ak can be expressedP as the fraction a (α + k)(α + k) · : : : · (α + k) k+1 = 1 2 p x: ak (k + 1)(β1 + k)(β2 + k) · : : : · (βq + k) If p = 2 and q = 1, we get Gauss's hypergeometric function 2F1(α, β; γ; x), which is the sum of the series αβ α(α + 1)β(β + 1) α(α + 1)(α + 2)β(β + 1)(β + 2) 1+ x+ x2+ x3+: : : : (2) 1 · γ 1 · 2 · γ(γ + 1) 1 · 2 · 3 · γ(γ + 1)(γ + 2) Gauss dealt with that series in [2]. He stated a condition for the convergence of (2) in the terms of the coefficients α, β, γ, x, which is described in the following theorem. z y y y yuw w ¨{{E ¨aa ~, RP {~ _ Theorem Let x 2 C and γ 2 Cnf0; −1; −2; −3; : : :g. If jxj < 1, then the series (2) is convergent. If jxj = 1, then the series (2) is convergent if and only if jγ − α − βj > 0 holds. In case of jxj > 1, the series (2) is divergent. "[0`ge fP^T]E\00d0cE^Te \0fJ^T\ ^T[0`^T[0`0\0]Ehg\Pj"^T[0`0pqkJkJp/jTd0f0cE^Te \0f Gauss derived a following formula for hypergeometric series (γ − α)(γ − β) F (α, β; γ; 1) = F (α, β; γ + 1; 1): (3) γ(γ − α − β) Then he generalized equation (3) into the form F (α, β; γ; 1) (γ − α)(γ + 1 − α) · : : : · (γ + k − 1 − α) = (4) γ(γ + 1) · : : : · (γ + k − 1) (γ − β)(γ + 1 − β) · : : : · (γ + k − 1 − β) × F (α, β; γ + k; 1); (γ − α − β)(γ + 1 − α − β) · : : : · (γ + k − 1 − α − β) where k 2 N. This recurrence formula became a starting point for his next investigation about the gama function. For k 2 N, z 2 C, Gauss introduced the function (k; z) by Q 1 · 2 · 3 · : : : · k (k; z) = kz: z z z : : : z k Y ( + 1)( + 2)( + 3) · · ( + ) A simple calculation shows that the expression (k; γ − 1) · (k; γ − α − β − 1) Q(k; γ − α −Q1) · (k; γ − β − 1) Q Q can be reduced to the fraction (γ − α)(γ + 1 − α) · : : : · (γ + k − 1 − α) γ(γ + 1) · : : : · (γ + k − 1) (γ − β)(γ + 1 − β) · : : : · (γ + k − 1 − β) × : (γ − α − β)(γ + 1 − α − β) · : : : · (γ + k − 1 − α − β) It means that the equation (4) can be transformed to the form (k; γ − 1) · (k; γ − α − β − 1) F (α, β; γ; 1) = · F (α, β; γ + k; 1): Q(k; γ − α −Q1) · (k; γ − β − 1) Q Q uwxwy{zP| }_~0"@ It is easily seen that (k; z) is defined for all z 2 C except the negative integers. If z is a nonnegativQ e integer, we get (for all k 2 N) (k; 0) = 1; Y 1 · k 1 (k; 1) = = ; k + 1 1 Y 1 + k 1 · 2 · k2 1 · 2 (k; 2) = = ; (k + 1)(k + 2) 1 2 Y 1 + k 1 + k . . 1 · 2 · 3 · : : : · z (k; z) = : 1 2 3 : : : z Y 1 + k 1 + k 1 + k · · 1 + k Gauss determined the values of the function (k; z + 1) for given k and z by the recurrence formula Q z + 1 (k; z + 1) = (k; z) · z+1 : Y Y 1 + k In a similar way, Gauss found a recurrence formula with respect to k z+1 1 + 1 (k + 1; z) = (k; z) · k : (5) 1+ z Y Y 1 + k The formula (5) yields the following equalities 1 (1; z) = ; z Y 1 + z+1 1 2 1 2z+1 (2; z) = · 1 = · ; 1 + z 2+ z 1 + z 2 + z Y 1 . 1 2z+1 3z+1 kz+1 (k; z) = · · · : : : · : z z z z z k z k z Y + 1 1 · (2 + ) 2 · (3 + ) ( − 1) · ( + ) Then Gauss supposed k ! 1, z as fixed point and set k = h, z < h. If h increase to h + 1, then the value of log (k; z) increase too. It holds by (5) Q z+1 z 1 + 1 1 + 1 log (h + 1; z) − log (h; z) = log h = log h : (6) 1+ z 1 + z Y Y 1 + h h+1 h+1 1 According to the equality = h and the equation (6), Gauss got h h+1 1 z log (h + 1; z) − log (h; z) = −z log 1 − − log 1 + : (7) h h Y Y + 1 + 1 z y y y yuw w ¨{{E ¨aa ~, RP {~ _ Both logarithms on the right-hand side of (7) can be expressed in the form of power series 1 1 z z2 z + + : : : − − + : : : : h + 1 2(h + 1)2 h + 1 2(h + 1)2 Assuming the absolute convergence of both series, Gauss got for the increase of log (h; z) the convergent series Q z(1 + z) z(1 − z2) z(1 + z3) z(1 − z4) + + + + : : : : : 2(h + 1)2 3(h + 1)3 4(h + 1)4 5(h + 1)5 If the value k increases from h + 1 to h + 2, then one has z(1 + z) z(1 − z2) z(1 + z3) log (h + 2; z) − log (h + 1; z) = + + + : : : : : h 2 h 3 h 4 Y Y 2( + 2) 3( + 2) 4( + 2) Generally, if the value k increases from h to h+n, then the difference log (h+ n; z) − log (h; z) equals Q 1 Q 1 1 1 1 z(1 + z) + + + : : : + 2 (h + 1)2 (h + 2)2 (h + 3)2 (h + n)2 1 1 1 1 1 + z(1 − z2) + + + : : : + 3 (h + 1)3 (h + 2)3 (h + 3)3 (h + n)3 1 1 1 1 1 + z(1 + z3) + + + : : : + + : : : : 4 (h + 1)4 (h + 2)4 (h + 3)4 (h + n)4 For n ! 1 Gauss obtained an absolute convergent double series 1 1 z + z2i z − z2i+1 + : 2i(h + j)2i (2i + 1)(h + j)2i+1 Xi=1 Xj=1 Gauss proved the finitness of limk!1 (k; z) for every C\{−1; −2; −3; : : :g.
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