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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> ...

==Differential equations== ==Delay Differential Equations== ...

...''F''₁ 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== ...

[[Category:Partial differential equations]] ...

...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}}: ...

...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 ...

...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 ...

...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 ...

..., 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 ...

...hebyshev polynomials solve the [[Chebyshev equation|Chebyshev differential equations]] ...ebyshev polynomials is as the solutions to [[Sturm–Liouville problem|those equations]].) ...

...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 == ...

...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} ...

...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 ...

...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 ...

...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] ...

...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. ...