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Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D In mathematics , a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n .
A trigonometric polynomial can be considered a periodic function on the real line, with period some divisor of , or as a function on the unit circle. Trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm; [4] this is a special case of the Stone–Weierstrass theorem.
Vincenty's goal was to express existing algorithms for geodesics on an ellipsoid in a form that minimized the program length (Vincenty 1975a). His unpublished report (1975b) mentions the use of a Wang 720 desk calculator, which had only a few kilobytes of memory. To obtain good accuracy for long lines, the solution uses the classical solution ...
The left-hand side of the equation is the generating function for the Legendre polynomials. As an example, the electric potential Φ( r , θ ) (in spherical coordinates ) due to a point charge located on the z -axis at z = a (see diagram right) varies as Φ ( r , θ ) ∝ 1 R = 1 r 2 + a 2 − 2 a r cos θ . {\displaystyle \Phi (r,\theta ...
Plot of the Jacobi polynomial function (,) with = and = and = in the complex plane from to + with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics , Jacobi polynomials (occasionally called hypergeometric polynomials ) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are a class of classical orthogonal ...
The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions.Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point.
The equation of a line: Ax + By = C, with A 2 + B 2 = 1 and C ≥ 0; The equation of a circle: () + = By contrast, there are alternative forms for writing equations. For example, the equation of a line may be written as a linear equation in point-slope and slope-intercept form.
Unlike Kummer's function which is an entire function of z, U(z) usually has a singularity at zero. For example, if b = 0 and a ≠ 0 then Γ(a+1)U(a, b, z) − 1 is asymptotic to az ln z as z goes to zero. But see #Special cases for some examples where it is an entire function (polynomial).