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  • The '''incomplete Bessel functions''' are defined as the same [[delay differential equation]]s of the complete-type [[Bessel functions]]: :<math>J_{v-1}(z,w)-J_{v+1}(z,w)=2\dfrac{\partial}{\partial z}J_v(z,w)</math> ...
    7 KB (1,416 words) - 19:17, 28 August 2020
  • ==Differential equations== ==Delay Differential Equations== ...
    4 KB (740 words) - 07:07, 20 December 2019
  • ...''F''<sub>1</sub> of one variable. Appell established the set of [[partial differential equation]]s of which these [[function (mathematics)|function]]s are solutio ==Derivatives and differential equations== ...
    15 KB (2,536 words) - 13:44, 16 September 2020
  • [[Category:Partial differential equations]] ...
    2 KB (225 words) - 20:59, 10 March 2014
  • ...ylinder functions''' are [[special function]]s defined as solutions to the differential equation ...y [[completing the square]] and rescaling ''z'', called [[H. F. Weber]]'s equations {{harv|Weber|1869}}: ...
    5 KB (794 words) - 13:06, 25 March 2019
  • ...erential equations|ordinary]] [[complex differential equation|differential equations in the complex plane]] with the '''Painlevé property''' (the only movable s ...are [[pole (complex analysis)|poles]]. This property is rare in nonlinear equations. Poincaré and L. Fuchs showed that any first order equation with the Painl ...
    18 KB (2,538 words) - 03:25, 20 May 2020
  • ...tion''', is a solution of '''Lamé's equation''', a second-order [[ordinary differential equation]]. It was introduced in the paper {{harvs|first=Gabriel|last= Lam ...olutions—called [[periodic instantons]], bounces or bubbles—of Schrödinger equations for various periodic and anharmonic potentials.<ref>H. J. W. Müller-Kirsten ...
    9 KB (1,317 words) - 21:17, 1 November 2020
  • ...ular, it occurs when solving [[Laplace's equation]] (and related [[partial differential equation]]s) in [[spherical coordinates]]. Associated Legendre polynomials ...it. The functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and ''m'' follows by ...
    30 KB (4,715 words) - 03:36, 23 October 2020
  • ..., sometimes called angular Mathieu functions, are solutions of Mathieu's [[differential equation]] ...cur in problems involving periodic motion, or in the analysis of [[partial differential equation]] [[boundary value problem]]s possessing [[elliptic]]{{Disambiguat ...
    41 KB (6,390 words) - 15:44, 19 November 2020
  • ...hebyshev polynomials solve the [[Chebyshev equation|Chebyshev differential equations]] ...ebyshev polynomials is as the solutions to [[Sturm–Liouville problem|those equations]].) ...
    49 KB (7,889 words) - 18:57, 19 November 2020
  • ...h>'s is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the c == Definition via differential equation == ...
    27 KB (4,247 words) - 00:26, 24 September 2020
  • ...h.org/encyclopedia/RigidityTheoremForAnalyticFunctions.html], stating that equations between holomorphic functions valid on a real interval, hold everywhere. In \frac{\partial \Gamma (s,x) }{\partial x} = - x^{s-1} e^{-x} ...
    37 KB (5,772 words) - 10:58, 16 November 2020
  • ...ted by {{math|''τ'' ↦ ''τ'' + 1}} and {{math|''τ'' ↦ −{{sfrac|1|''τ''}}}}. Equations for the first transform are easily found since adding one to {{mvar|τ}} in ...y conditions.<ref>{{Cite journal|last=Ohyama|first=Yousuke|date=1995|title=Differential relations of theta functions|url=https://projecteuclid.org/euclid.ojm/12007 ...
    28 KB (4,239 words) - 05:08, 1 October 2020
  • ...nction that occurs often in [[probability]], [[statistics]], and [[partial differential equation]]s. In many of these applications, the function argument is a real ...[[Closed-form expression|closed form]] in terms of [[Elementary function (differential algebra)|elementary functions]], but by expanding the [[integrand]] ''e''<s ...
    39 KB (5,667 words) - 15:45, 4 November 2020
  • ...IXYC&pg=PA28 Extract of page 28]</ref><ref>{{cite book |title=Differential Equations: An Introduction with Mathematica |edition=illustrated |first1=Clay C. |las Other important functional equations for the gamma function are [[reflection formula|Euler's reflection formula] ...
    72 KB (11,210 words) - 03:19, 5 November 2020
  • ...form parameters can be obtained from the general form coefficients by the equations: ...tic]] of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae. ...
    76 KB (12,397 words) - 06:11, 3 November 2020