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1: Customize DLMF
Customize DLMF
You can customize the appearance and functionality of the DLMF site using these selections. Note: Javascript and Cookies must be enabled to support customizations.2: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►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). …3: About MathML
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►, built-in to the browser) support for MathML is growing, (see Browsers supporting MathML).
…By default, DLMF will use Native support when available; You may choose how MathML is processed (Native or MathJax) at Customize DLMF.
►In rare cases, a browser lacks both MathML support and a robust enough javascript implementation capable of running MathJax; you may wish to visit the Customize DLMF page and choose the HTML+images document format.
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Browsers supporting MathML
… ►Recent enhancements to the WebKit engine now provide support for MathML Core. …4: Browsers
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►Most modern browsers support either MathML, or the MathJax fallback acceptably; If yours does not, please consider upgrading.
…If none of those solutions work for you, you may explicitly choose a format such as HTML+images, using images for mathematics, at Customize DLMF.
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5: 33.14 Definitions and Basic Properties
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►Again, there is a regular singularity at with indices and , and an irregular singularity of rank 1 at .
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►The functions and are defined by
…An alternative formula for is
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►Note that the functions , , do not form a complete orthonormal system.
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►With arguments suppressed,
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6: Viewing DLMF Interactive 3D Graphics
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►WebGL is supported in the current versions of most common web browsers.
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►1, some advanced features of X3DOM are currently not fully supported (see x3dom.org).
…If you have trouble viewing the WebGL visualizations in your web browser, see x3dom.org or caniuse.com/webgl for information on WebGL browser support.
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►After installing the viewer you must select Customize DLMF on the DLMF Menu bar and choose either VRML or X3D under “Visualization Format.
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►Please see caniuse.com/webgl or x3dom.org for information on WebGL browser support.
7: 33.9 Expansions in Series of Bessel Functions
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►where the function is as in §10.47(ii), , , and
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33.9.3
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►Here , , and
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33.9.5
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►For other asymptotic expansions of see Fröberg (1955, §8) and Humblet (1985).