Special Functions||(Important Results) || Gate Net TIFR Jest 2026 скачать в хорошем качестве

Special Functions||(Important Results) || Gate Net TIFR Jest 2026

The video provides an overview of special functions relevant for exams like GATE, NET, TIFR, and JEST 2026. It covers several types of polynomials and functions, explaining their differential equations, general expressions, important results, and properties. Here's a breakdown of the key topics: Hermite Polynomials (0:04-4:32): Defined as solutions to the Hermite differential equation (0:14-0:21). Discusses properties like Hn(-x), values at x=0, derivatives, and the range of their nodes (-infinity to infinity) (1:23-2:23). Explains the generating function (2:26-3:02) and orthogonality condition (3:09-4:01), noting the multiplication by e^(-x^2) for the latter. Bessel Functions (4:35-9:48): Introduces the Bessel differential equation (4:39-4:58). Highlights that Bessel functions can be expressed in series form similar to sine and cosine, but with different coefficients (5:28-6:11). Explains the graphical trends for even and odd 'n' values (6:14-6:54). Covers important results including jn(-x), j(-n)(x), and recurrence relations (7:38-8:24). Discusses the generating function for Bessel functions (8:29-8:44). Briefly touches upon finding positive and negative real roots of polynomials (8:47-9:47). Laguerre Polynomials (9:51-11:10): Presents the Laguerre differential equation (9:53-10:02). Notes that Laguerre polynomials are neither odd nor even functions (10:22-10:26). Mentions that their nodes lie between 0 and infinity (10:28-10:39). Explains the generating function (10:51-10:55) and orthogonality condition (10:58-11:08). Gamma and Beta Functions (11:10-13:02): Introduces the Gamma function and its definition (11:14-11:26). Discusses various important results and properties of the Gamma function, including factorial relations and expansions (11:28-11:58). Defines the Beta function with its integral form (12:01-12:22). Explores properties of the Beta function, such as symmetry and its relationship with the Gamma function (12:30-13:02).