Euler–Poisson–Darboux equation
(0.004 seconds)
1—10 of 622 matching pages
1: 24.1 Special Notation
…
►
Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 30.2 Differential Equations
…
►
§30.2(i) Spheroidal Differential Equation
… ►The equation contains three real parameters , , and . … ►With Equation (30.2.1) changes to … ►If , Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …If , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).3: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
►§15.10(i) Fundamental Solutions
►
15.10.1
►This is the hypergeometric differential equation.
…
►
…
4: 31.2 Differential Equations
…
►
§31.2(i) Heun’s Equation
►
31.2.1
.
►This equation has regular singularities at , with corresponding exponents , , , , respectively (§2.7(i)).
…
►The parameters play different roles: is the singularity parameter; are exponent parameters; is the accessory parameter.
…
►
satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters ; , , .
…
5: 29.2 Differential Equations
§29.2 Differential Equations
►§29.2(i) Lamé’s Equation
… ►§29.2(ii) Other Forms
… ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1).6: 32.2 Differential Equations
…
►The six Painlevé equations
– are as follows:
…
►with , , , and arbitrary constants.
…
►For arbitrary values of the parameters , , , and , the general solutions of – are transcendental, that is, they cannot be expressed in closed-form elementary functions.
…
►If in , then set and , without loss of generality, by rescaling and if necessary.
…Lastly, if and , then set and , without loss of generality.
…
7: 28.2 Definitions and Basic Properties
…
►
§28.2(i) Mathieu’s Equation
… ►
28.2.1
…
►This is the characteristic equation of Mathieu’s equation (28.2.1).
…
►
§28.2(iv) Floquet Solutions
… ► …8: 28.20 Definitions and Basic Properties
…
►
§28.20(i) Modified Mathieu’s Equation
►When is replaced by , (28.2.1) becomes the modified Mathieu’s equation: ►
28.20.1
…
►
28.20.2
.
…
►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant.
…
9: 19.18 Derivatives and Differential Equations
…
►and two similar equations obtained by permuting in (19.18.10).
►More concisely, if , then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation:
…and also a system of
Euler–Poisson differential equations (of which only are independent):
…
►The next four differential equations apply to the complete case of and in the form (see (19.16.20) and (19.16.23)).
►The function satisfies an Euler–Poisson–Darboux equation:
…