L’Hôpital rule
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1: 25.15 Dirichlet -functions
§25.15 Dirichlet -functions
►§25.15(i) Definitions and Basic Properties
►The notation was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … … ►§25.15(ii) Zeros
…2: 3.5 Quadrature
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§3.5(i) Trapezoidal Rules
… ►The composite trapezoidal rule is … ►§3.5(ii) Simpson’s Rule
… ►§3.5(iv) Interpolatory Quadrature Rules
… ► …3: 1.4 Calculus of One Variable
4: 3.2 Linear Algebra
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►With the process of solution can then be regarded as first solving the equation for (forward
elimination), followed by the solution of for (back substitution).
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►Because of rounding errors, the residual vector
is nonzero as a rule.
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3.2.8
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►In the case that the orthogonality condition is replaced by -orthogonality, that is, , , for some positive definite matrix with Cholesky decomposition , then the details change as follows.
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5: 34.5 Basic Properties: Symbol
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►Equations (34.5.15) and (34.5.16) are the sum rules.
They constitute addition theorems for the symbol.
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34.5.11
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6: 18.4 Graphics
7: 23.10 Addition Theorems and Other Identities
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23.10.10
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23.10.17
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23.10.18
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23.10.19
►Also, when is replaced by the lattice invariants and are divided by and , respectively.
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8: 18.36 Miscellaneous Polynomials
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§18.36(v) Non-Classical Laguerre Polynomials ,
… ►For the Laguerre polynomials this requires, omitting all strictly positive factors, … ►implying that, for , the orthogonality of the with respect to the Laguerre weight function , . …These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the polynomials, self-adjointness implying both orthogonality and completeness. … ►The resulting EOP’s, , satisfy …9: 23.14 Integrals
10: 23.2 Definitions and Periodic Properties
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►The generators of a given lattice are not unique.
…where are integers, then , are generators of iff
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►When the functions are related by
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►When it is important to display the lattice with the functions they are denoted by , , and , respectively.
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►If , is any pair of generators of , and is defined by (23.2.1), then
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