In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation
| (1) |
This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates.
The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling z, called H. F. Weber's equations: [1]
| (A) |
and
| (B) |
If
is a solution, then so are
If
is a solution of equation ( A ), then
is a solution of ( B ), and, by symmetry,
are also solutions of ( B ).
There are independent even and odd solutions of the form ( A ). These are given by (following the notation of Abramowitz and Stegun (1965)): [2]
and
where is the confluent hypergeometric function.
Other pairs of independent solutions may be formed from linear combinations of the above solutions. [2] One such pair is based upon their behavior at infinity:
where
The function U(a, z) approaches zero for large values of z and |arg(z)| < π/2, while V(a, z) diverges for large values of positive real z .
and
For half-integer values of a, these (that is, U and V) can be re-expressed in terms of Hermite polynomials; alternatively, they can also be expressed in terms of Bessel functions.
The functions U and V can also be related to the functions Dp(x) (a notation dating back to Whittaker (1902)) [3] that are themselves sometimes called parabolic cylinder functions: [2]
Function Da(z) was introduced by Whittaker and Watson as a solution of eq.~( 1 ) with bounded at . [4] It can be expressed in terms of confluent hypergeometric functions as
Power series for this function have been obtained by Abadir (1993). [5]
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation; it is indeed derived using the product rule.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
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In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
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In mathematics, the Struve functionsHα(x), are solutions y(x) of the non-homogeneous Bessel's differential equation:
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The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The normal distribution is recovered as q → 1.