# 오일러-가우스 초기하함수2F1

둘러보기로 가기 검색하러 가기

## 개요

• 초기하급수$\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1$

여기서 $$(a)_n=a(a+1)(a+2)...(a+n-1)$$에 대해서는 포흐하머 (Pochhammer) 기호 항목 참조

• 적분표현$\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt$
• 초기하급수의 해석적확장을 통해 얻어진 함수를 초기하함수라 함
• 오일러, 가우스, 쿰머, 리만,슈워츠 등의 연구

## 초기하 미분방정식

• $$w(z)=\,_2F_1(a,b;c;z)$$ 는 다음 피카드-Fuchs 형태의 미분방정식의 해가 된다

$z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0$

## 오일러의 변환 공식

$$_2F_1 (a,b;c;z) = (1-z)^{-a} {}_2F_1 (a, c-b;c ; \frac{z}{z-1})$$

$$_2F_1 (a,b;c;z) = (1-z)^{-b}{}_2F_1(c-a,b;c;\frac{z}{z-1})$$

$$_2F_1 (a,b;c;z) = (1-z)^{c-a-b}{}_2F_1 (c-a, c-b;c ; z)$$

증명

다음 적분표현을 활용

$$\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt$$

위의 우변에서 $$t\to 1-t$$, $$t\to \frac{t}{1-z-tz}$$, $$t\to \frac{1-t}{1-tz}$$의 변환을 이용하면 항등식이 얻어진다. ■

## special values

• Chu-Vandermonde 공식$\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}$ 아래 가우스 공식에서 $$a=-n$$인 경우에 얻어진다
• 가우스 공식$\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$
• 위의 두 식에 대해서는 초기하 급수의 합공식
• 렘니스케이트(lemniscate) 곡선과 타원적분$\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;\frac{1}{2})=K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots$
• http://mathworld.wolfram.com/HypergeometricFunction.html$_2F_1(\frac{1}{3},\frac{2}{3};\frac{5}{6};\frac{27}{32})=\frac{8}{5}$$_2F_1(\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{80}{81})=\frac{9}{5}$$_2F_1(\frac{1}{8},\frac{3}{8};\frac{1}{2};\frac{2400}{2401})=\frac{2}{3}\sqrt{7}$$_2F_1(\frac{1}{6},\frac{1}{3};\frac{1}{2};\frac{25}{27})=\frac{3}{4}\sqrt{3}$$_2F_1(\frac{1}{6},\frac{1}{2};\frac{2}{3};\frac{125}{128})=\frac{4}{3}\sqrt[6]2$$_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{1323}{1331})=\frac{3}{4}\sqrt[4]{11}$$_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{121}{125})=\frac{\sqrt[6]{2}\sqrt[4]{15}}{4\sqrt{\pi}}\frac{\Gamma(\frac{1}{3})^3}{\Gamma(\frac{1}{4})^2}(1+\sqrt{3})$

## 노트

### 말뭉치

1. This paper introduces the hypergeo package of R routines, for numerical calculation of hypergeometric functions.[1]
2. The package is focussed on ecient and accurate evaluation of the hypergeometric function over the whole of the complex plane within the constraints of xed-precision arithmetic.[1]
3. 2 Numerical evaluation of the Gauss hypergeometric function with the hypergeo package when dened.[1]
4. Writing a, b, c for the two upper and one lower argument respectively, the resulting function 2F1 (a, b; c; z) is known as the hypergeometric function.[1]
5. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al.[2]
6. Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1/2, 1/3, ... or 0.[2]
7. We give a basic introduction to the properties of Gauss hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hyperegeo- metric equation.[3]
8. Initially this document started as an informal introduction to Gauss hypergeometric functions for those who want to have a quick idea of some main facts on hypergeometric functions.[3]
9. To confuse matters even more, the term "hypergeometric function" is less commonly used to mean closed form, and "hypergeometric series" is sometimes used to mean hypergeometric function.[4]
10. The hypergeometric functions are solutions to the hypergeometric differential equation, which has a regular singular point at the origin.[4]
11. Many functions of mathematical physics can be expressed as special cases of the hypergeometric functions.[4]
12. We consider the asymptotic behavior of the Gauss hypergeometric function when several of the parameters a,b,c are large.[5]
13. Computes the Gauss hypergeometric function 2F1(a,b;c;z) and its derivative for real z, z<1 by integrating the defining differential equation using the Matlab differential equation solver ode15i.[6]
14. The major development of the theory of hypergeometric function was carried out by Gauss and published in his famous work of 1812.[7]
15. Almost all of the elementary functions of Mathematics are either hypergeometric, ratios of hypergeometric functions or limiting cases of a hypergeometric series.[7]
16. Two hypergeometric functions with the same argument z are contiguous if their parameters a, b and c differ by integers.[7]
17. A contiguous relation between any three contiguous hypergeometric functions can be found by combining linearly a sequence of Gauss contiguous relations.[7]
18. In this course we will study multivariate hypergeometric functions in the sense of Gelfand, Kapranov, and Zelevinsky (GKZ systems).[8]
19. These functions generalize the classical hypergeometric functions of Gauss, Horn, Appell, and Lauricella.[8]
20. We end with a brief discussion of the classication problem for rational hypergeometric functions.[8]
21. For one-variable hypergeometric functions this interplay has been well understood for several decades.[8]
22. Abstract The two most commonly used hypergeometric functions are the conflu- ent hypergeometric function and the Gauss hypergeometric function.[9]
23. Except for specific situations, computing hypergeometric functions is difficult in practice.[9]
24. (a)j (b)j zj j ! , which is also commonly denoted by M(a; b; z) and is itself often called the confluent hypergeometric function.[9]
25. The function 2F1(a, b; c; z) is commonly denoted by F (a, b; c; z) and is also fre- quently called the Gauss hypergeometric function.[9]
26. We nd that, for both the conuent and Gauss hypergeometric functions, there is no simple answer to the problem of their computation, and dierent methods are optimal for dierent parameter regimes.[10]
27. 3.3 Writing the conuent hypergeometric function as a single fraction . . . . . .[10]
28. 4.3 Writing the Gauss hypergeometric function as a single fraction .[10]
29. The computation of the hypergeometric function pFq, a special function encountered in a variety of applications, is frequently sought.[10]
30. Returns hyp2f1 scalar or ndarray The values of the gaussian hypergeometric function.[11]
31. Using (41) and subordination property, we have thatBy adopting (13) and (64), we obtainApplying hypergeometric function (59), we obtain the upper bound for the case .[12]
32. Recently, there has been a clear interest on Bessel and hypergeometric functions from the point of view of geometric function theory.[12]
33. A hypergeometric function is the sum of a hypergeometric series, which is dened as follows.[13]
34. When one of the numerator parameters ai equals N , where N is a nonnegative integer, the hypergeometric function is a polynomial in z (see below).[13]
35. Sometimes the most general hypergeometric function pFq is called a generalized hypergeo- metric function.[13]
36. For the hypergeometric function 2F1 we have an integral representation due to Euler: Theorem 1.[13]
37. This generally speeds up evaluation by producing a hypergeometric function of lower order.[14]
38. Euler introduced the power series expansion of the form: where a, b, c are rational functions and F(a, b, c, z) is called the hypergeometric function.[15]
39. The hypergeometric function takes a prominent position amongst the world of standard mathematical functions used in both pure and applied mathematics.[15]
40. Gauss was aware of the multi-valuedness of the hypergeometric functions, known in recent times as the monodromy problem.[15]
41. The modern notation for the Gauss hypergeometric function is according to Barnes (1908).[15]
42. The classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.[16]
43. This paper claims that the natural logarithm can be represented by the Gaussian hypergeometric function.[17]
44. We establish two quadratic transformations for Gauss hypergeometric function in terms of nite summation of combination of two Clausen hypergeometric functions.[18]
45. Further, we have generalized our quadratic transformations in terms of general double series identities as well as in terms of reduction formulas for Kamp de Friets double hypergeometric function.[18]
46. The Hypergeometric functions of two variables was introduced by Appell (1880) and Lauricella (1893) generalized them to several variables.[18]
47. It is interesting to mention here that the results are very important for the application point of view, whenever hypergeometric functions reduce to gamma functions.[18]
48. In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functions .[19]
49. For , and , we have where the , a special case of the generalized hypergeometric function (10), is the Gauss hypergeometric function.[19]
50. Further, if we set and in Theorems 1 to 5 or make use of the result (8), we obtain various integral transforms and fractional integral formulas for the Gauss hypergeometric function .[19]
51. This a hypergeometric equation with constants a, b and c dened by F = c, G = (a + b + 1) and H = ab and can therefore be solved near t = 0 and t = 1 in terms of the hypergeometric function.[20]
52. Problems: Find the general solution of each of the following dierential equations near the indicated singular point in terms of hypergeometric function.[20]
53. However, the hypergeometric function is defined over the whole of the complex plane, so analytic continuation may be used if appropriate cut lines are used.[21]
54. Gausss hypergeometric function gives a modular parame- terization of period integrals of elliptic curves in Legendre normal form E() : y2 = x(x 1)(x ).[22]
55. Legendre elliptic curves, hypergeometric functions.[22]
56. Hypergeometric functions are rarely in a form in which these formulae can be applied directly.[23]
57. Writing sums as hypergeometric functions has the great advantage of simplifying manipulation by computer algebraic methods.[23]
58. It is assumed that the hypergeometric functions are convergent and do not contain negative integers in the bottom parameter list.[23]
59. A particular solution of Gausss hypergeometric differential equation (1) is known as Gausss hypergeometric function or simply hypergeometric function.[24]
60. 2 ​ F 1 ​ ( a , b , c , z ) Regularized Gauss hypergeometric function RisingFactorial ( z ) k \left(z\right)_{k} ( z ) k ​ Rising factorial Pow a b {a}^{b} a b Power Factorial n ! n ![25]
61. The hypergeometric functions are solutions to the Hypergeometric Differential Equation, which has a Regular Singular Point at the Origin.[26]

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'hypergeometric'}, {'LEMMA': 'function'}]
• [{'LOWER': 'gaussian'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'function'}]
• [{'LOWER': 'ordinary'}, {'LOWER': 'hypergeometric'}, {'LEMMA': 'function'}]