integer parameters
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11—20 of 276 matching pages
11: 28.26 Asymptotic Approximations for Large
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28.26.1
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : hyperbolic cosine function, : imaginary unit, : radial Mathieu function, : integer, : parameter, : complex variable and
- A&S Ref:
- 20.9.7 (in slightly different notation)
- Permalink:
- http://dlmf.nist.gov/28.26.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.26(i), §28.26 and Ch.28
28.26.2
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : hyperbolic cosine function, : imaginary unit, : radial Mathieu function, : integer, : parameter, : complex variable and
- A&S Ref:
- 20.9.7 (in slightly different notation)
- Permalink:
- http://dlmf.nist.gov/28.26.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.26(i), §28.26 and Ch.28
28.26.3
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ⓘ
- Symbols:
- : hyperbolic sine function, : arctangent function, : integer, : parameter, : complex variable and
- A&S Ref:
- 20.9.8 (in slightly different notation)
- Permalink:
- http://dlmf.nist.gov/28.26.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.26(i), §28.26 and Ch.28
28.26.4
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ⓘ
- Symbols:
- : Poincaré asymptotic expansion, : hyperbolic cosine function, : integer, : parameter and : complex variable
- A&S Ref:
- 20.9.9 (in slightly different notation)
- Referenced by:
- §28.26(i)
- Permalink:
- http://dlmf.nist.gov/28.26.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.26(i), §28.26 and Ch.28
28.26.5
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- Symbols:
- : Poincaré asymptotic expansion, : hyperbolic cosine function, : hyperbolic sine function, : integer, : parameter and : complex variable
- A&S Ref:
- 20.9.10 (in slightly different notation)
- Referenced by:
- §28.26(i)
- Permalink:
- http://dlmf.nist.gov/28.26.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.26(i), §28.26 and Ch.28
12: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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31.5.1
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- Symbols:
- : real or complex parameter, : nonnegative integer, : complex parameter, : coefficient, : coefficient and : coefficient
- Permalink:
- http://dlmf.nist.gov/31.5.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.5 and Ch.31
31.5.2
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ⓘ
- Defines:
- : Heun polynomials
- Symbols:
- : Heun functions, : complex variable, : real or complex parameter, : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/31.5.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.5 and Ch.31
13: 29.7 Asymptotic Expansions
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29.7.1
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- Symbols:
- : eigenvalues of Lamé’s equation, : Poincaré asymptotic expansion, : nonnegative integer, : nonnegative integer, : real parameter, : real parameter, and : coefficients
- Permalink:
- http://dlmf.nist.gov/29.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.7(i), §29.7 and Ch.29
29.7.3
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ⓘ
- Symbols:
- : nonnegative integer, : real parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/29.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.7(i), §29.7 and Ch.29
29.7.4
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ⓘ
- Symbols:
- : nonnegative integer, : real parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/29.7.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.7(i), §29.7 and Ch.29
29.7.6
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ⓘ
- Symbols:
- : nonnegative integer, : real parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/29.7.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.7(i), §29.7 and Ch.29
29.7.7
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- Symbols:
- : nonnegative integer, : real parameter and : coefficients
- Permalink:
- http://dlmf.nist.gov/29.7.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.7(i), §29.7 and Ch.29
14: 28.10 Integral Equations
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28.10.1
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- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : integral, : parameter, : integer, : complex variable and : Fourier coefficient
- A&S Ref:
- 20.7.20 (in different form)
- Permalink:
- http://dlmf.nist.gov/28.10.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.10(i), §28.10 and Ch.28
28.10.2
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ⓘ
- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic cosine function, : integral, : sine function, : parameter, : integer, : complex variable and : Fourier coefficient
- A&S Ref:
- 20.7.21 (in different form)
- Permalink:
- http://dlmf.nist.gov/28.10.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.10(i), §28.10 and Ch.28
28.10.3
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ⓘ
- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : integral, : sine function, : parameter, : integer, : complex variable and : Fourier coefficient
- A&S Ref:
- 20.7.20 (in different form)
- Permalink:
- http://dlmf.nist.gov/28.10.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.10(i), §28.10 and Ch.28
28.10.5
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ⓘ
- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : differential of , : hyperbolic sine function, : integral, : sine function, : parameter, : integer, : complex variable and : Fourier coefficient
- A&S Ref:
- 20.7.23 (in different form)
- Permalink:
- http://dlmf.nist.gov/28.10.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.10(i), §28.10 and Ch.28
28.10.10
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ⓘ
- Symbols:
- : Bessel function of the first kind, : Mathieu function, : eigenvalues of Mathieu equation, : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : integral, : parameter, : integer and : fundamental solution
- Permalink:
- http://dlmf.nist.gov/28.10.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.10(ii), §28.10 and Ch.28
15: 28.2 Definitions and Basic Properties
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28.2.19
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- Symbols:
- : element of, : set of all integers, : parameter, : integer, : complex parameter, : parameter and : coefficients
- A&S Ref:
- 20.3.12 20.3.13
- Referenced by:
- §28.2(iv)
- Permalink:
- http://dlmf.nist.gov/28.2.E19
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.2(iv), §28.2 and Ch.28
28.2.21
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ⓘ
- Symbols:
- : eigenvalues of Mathieu equation, : the ratio of the circumference of a circle to its diameter, : parameter, : integer, : parameter, : Mathieu’s equation solution and : fundamental solution
- Referenced by:
- §28.2(vi), §28.29(ii), §28.7
- Permalink:
- http://dlmf.nist.gov/28.2.E21
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.2(v), §28.2 and Ch.28
28.2.22
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ⓘ
- Symbols:
- : eigenvalues of Mathieu equation, : the ratio of the circumference of a circle to its diameter, : parameter, : integer, : parameter, : Mathieu’s equation solution and : fundamental solution
- Referenced by:
- §28.2(vi), §28.29(ii), §28.7
- Permalink:
- http://dlmf.nist.gov/28.2.E22
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.2(v), §28.2 and Ch.28
28.2.31
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ⓘ
- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : integer, : parameter, : integer and : real variable
- Permalink:
- http://dlmf.nist.gov/28.2.E31
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.2(vi), §28.2 and Ch.28
28.2.34
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- Symbols:
- : Mathieu function, : the ratio of the circumference of a circle to its diameter, : parameter, : integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/28.2.E34
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.2(vi), §28.2 and Ch.28
16: 36.7 Zeros
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36.7.4
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : real parameter and : integer
- Permalink:
- http://dlmf.nist.gov/36.7.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §36.7(iii), §36.7 and Ch.36
36.7.6
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : cosine function, : exponential function, : imaginary unit, : real parameter, : real parameter, : integer and : real parameter
- Permalink:
- http://dlmf.nist.gov/36.7.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §36.7(iii), §36.7 and Ch.36
17: 31.15 Stieltjes Polynomials
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31.15.2
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- Symbols:
- : complex variable, : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter and : number of singularities
- Referenced by:
- §31.15(ii)
- Permalink:
- http://dlmf.nist.gov/31.15.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.15(ii), §31.15 and Ch.31
31.15.3
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- Symbols:
- : complex variable, : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : number of singularities and : zero of
- Permalink:
- http://dlmf.nist.gov/31.15.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.15(ii), §31.15 and Ch.31
31.15.6
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ⓘ
- Symbols:
- : nonnegative integer, : complex parameter and : number of singularities
- Referenced by:
- §31.15(iii)
- Permalink:
- http://dlmf.nist.gov/31.15.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.15(ii), §31.15 and Ch.31
31.15.7
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►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index , where each is a nonnegative integer, there is a unique Stieltjes polynomial with zeros in the open interval for each .
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- Symbols:
- : complex variable, : real or complex parameter, : nonnegative integer, : nonnegative integer, : complex parameter, : real or complex parameter and : number of singularities
- Permalink:
- http://dlmf.nist.gov/31.15.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §31.15(ii), §31.15 and Ch.31
18: 28.8 Asymptotic Expansions for Large
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28.8.4
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- Defines:
- : function (locally)
- Symbols:
- : parabolic cylinder function, : Poincaré asymptotic expansion, : binomial coefficient, : factorial (as in ), : integer, : parameter and : variable
- A&S Ref:
- 20.9.17 (in different form)
- Permalink:
- http://dlmf.nist.gov/28.8.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.8(ii), §28.8 and Ch.28
28.8.5
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ⓘ
- Defines:
- : function (locally)
- Symbols:
- : parabolic cylinder function, : Poincaré asymptotic expansion, : binomial coefficient, : factorial (as in ), : integer, : parameter and : variable
- A&S Ref:
- 20.9.18 (in slightly different form)
- Referenced by:
- Erratum (V1.0.16) for Equation (28.8.5)
- Permalink:
- http://dlmf.nist.gov/28.8.E5
- Encodings:
- pMML, png, TeX
- Error (effective with 1.0.16):
-
Originally the in front of the was given incorrectly as .
Suggested 2017-02-02 by Daniel Karlsson
- See also:
- Annotations for §28.8(ii), §28.8 and Ch.28
28.8.9
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ⓘ
- Defines:
- : function (locally)
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : cosine function, : base of natural logarithm, : sine function, : integer, : parameter and : real variable
- A&S Ref:
- 20.9.13 (in slightly different form)
- Permalink:
- http://dlmf.nist.gov/28.8.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.8(iii), §28.8 and Ch.28
28.8.11
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ⓘ
- Defines:
- (locally)
- Symbols:
- : Poincaré asymptotic expansion, : cosine function, : integer, : parameter and : real variable
- A&S Ref:
- 20.9.14 (in different form)
- Permalink:
- http://dlmf.nist.gov/28.8.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.8(iii), §28.8 and Ch.28
28.8.12
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- Defines:
- (locally)
- Symbols:
- : Poincaré asymptotic expansion, : cosine function, : sine function, : integer, : parameter and : real variable
- A&S Ref:
- 20.9.14 (in different form)
- Permalink:
- http://dlmf.nist.gov/28.8.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §28.8(iii), §28.8 and Ch.28
19: 24.17 Mathematical Applications
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►Let and , and be integers such that , , and is absolutely integrable over .
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24.17.1
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- Symbols:
- : Euler polynomials, : factorial (as in ), : integer, : integer, : integer, : parameter, : integer, : integer, : integrable function and : remainder
- Permalink:
- http://dlmf.nist.gov/24.17.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24
24.17.2
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- Symbols:
- : differential of , : factorial (as in ), : integral, : periodic Euler functions, : integer, : real or complex, : parameter, : integer, : integer, : integrable function and : remainder
- Permalink:
- http://dlmf.nist.gov/24.17.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24
20: 8.18 Asymptotic Expansions of
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8.18.1
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- Symbols:
- : order not exceeding, : gamma function, : incomplete beta function, : nonnegative integer, : nonnegative integer, : parameter, : parameter and : variable
- Referenced by:
- §8.18(i)
- Permalink:
- http://dlmf.nist.gov/8.18.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.18(i), §8.18 and Ch.8
8.18.3
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ⓘ
- Symbols:
- : order not exceeding, : gamma function, : incomplete beta function, : Poincaré asymptotic expansion, : nonnegative integer, : nonnegative integer, : parameter, : parameter, : variable, : functions and : coefficients
- Referenced by:
- Erratum (V1.0.14) for Equation (8.18.3)
- Permalink:
- http://dlmf.nist.gov/8.18.E3
- Encodings:
- pMML, png, TeX
- Addition (effective with 1.0.14):
- Previously this equation appeared without the order estimate as . The range of was extended to include .
- See also:
- Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8
8.18.4
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- Symbols:
- : nonnegative integer, : parameter, : parameter, : change of variable and : functions
- Permalink:
- http://dlmf.nist.gov/8.18.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8
8.18.6
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ⓘ
- Symbols:
- : base of natural logarithm, : nonnegative integer, : parameter, : change of variable and : coefficients
- Permalink:
- http://dlmf.nist.gov/8.18.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8
8.18.9
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- Symbols:
- : incomplete beta function, : Poincaré asymptotic expansion, : the ratio of the circumference of a circle to its diameter, : complementary error function, : nonnegative integer, : parameter, : parameter, : variable and : coefficients
- Permalink:
- http://dlmf.nist.gov/8.18.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8