2F1funktio

2F1funktio, also known as the Gauss or ordinary hypergeometric function, is denoted as 2F1(a,b;c;z). It is defined by the power series 2F1(a,b;c;z) = sum_{n=0}^∞ (a)_n (b)_n / [(c)_n n!] z^n, where (q)_n is the rising Pochhammer symbol. The series converges for |z| < 1 and can be analytically continued beyond this disk. The function is symmetric in a and b.

The 2F1funktio satisfies the hypergeometric differential equation: z(1−z) w'' + [c − (a+b+1) z] w' − ab w = 0, with regular singularities at z = 0, 1, and ∞. This equation makes the function a central object in problems with three regular singular points.

Integral representations are available, for example the Euler-type integral: 2F1(a,b;c;z) = Γ(c) / [Γ(b) Γ(c−b)] ∫_0^1 t^{b−1} (1−t)^{c−b−1} (1−z t)^{−a} dt, valid for Re c > Re b > 0. There are several transformation and contiguous relations, including Euler’s transformation 2F1(a,b;c;z) = (1−z)^{c−a−b} 2F1(c−a, c−b; c; z) and Pfaff’s transformation 2F1(a,b;c;z) = (1−z)^{−a} 2F1(a, c−b; c; z/(z−1)).

Special cases include termination to a polynomial when a or b is a nonpositive integer, and values at z = 1 given by Gauss’ summation formula when Re(c−a−b) > 0: 2F1(a,b;c;1) = Γ(c) Γ(c−a−b) / [Γ(c−a) Γ(c−b)]. The function plays a key role in mathematics and physics, appearing in differential equations, probability, and spectral theory.