change of variables
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11—20 of 93 matching pages
11: 9.6 Relations to Other Functions
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9.6.1
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9.6.10
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ⓘ
- Defines:
- : change of variable (locally)
- Symbols:
- : complex variable
- Proof sketch:
- Derivable from those given in Olver (1997b, pp. 392–393).
- Permalink:
- http://dlmf.nist.gov/9.6.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.6(ii), §9.6 and Ch.9
9.6.11
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ⓘ
- Symbols:
- : Airy function, : Airy function, : Bessel function of the first kind, : complex variable and : change of variable
- Proof sketch:
- Derivable from those given in Olver (1997b, pp. 392–393).
- A&S Ref:
- 10.4.22
- Permalink:
- http://dlmf.nist.gov/9.6.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.6(ii), §9.6 and Ch.9
9.6.12
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ⓘ
- Symbols:
- : Airy function, : Airy function, : Bessel function of the first kind, : complex variable and : change of variable
- Proof sketch:
- Derivable from those given in Olver (1997b, pp. 392–393).
- A&S Ref:
- 10.4.27
- Permalink:
- http://dlmf.nist.gov/9.6.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.6(ii), §9.6 and Ch.9
9.6.13
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- Symbols:
- : Airy function, : Airy function, : modified Bessel function of the first kind, : complex variable and : change of variable
- Proof sketch:
- Derivable from those given in Olver (1997b, pp. 392–393).
- A&S Ref:
- 10.4.25
- Permalink:
- http://dlmf.nist.gov/9.6.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.6(ii), §9.6 and Ch.9
12: 22.9 Cyclic Identities
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22.9.7
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- Defines:
- : change of variable (locally)
- Symbols:
- : Jacobian elliptic function, : Legendre’s complete elliptic integral of the first kind and : modulus
- Referenced by:
- §22.9(iii), §22.9(iv)
- Permalink:
- http://dlmf.nist.gov/22.9.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22
22.9.8
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ⓘ
- Symbols:
- : modulus, : cyclic quantity and : change of variable
- Referenced by:
- Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)
- Permalink:
- http://dlmf.nist.gov/22.9.E8
- Encodings:
- pMML, png, TeX
- Correction (effective with 1.0.25):
-
Originally this equation was written incorrectly for all
with . It has been corrected by writing throughout.
Suggested 2019-11-07 by Juan Miguel Nieto
- See also:
- Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22
22.9.13
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- Symbols:
- : cyclic quantity and : change of variable
- Permalink:
- http://dlmf.nist.gov/22.9.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22
22.9.14
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ⓘ
- Symbols:
- : cyclic quantity and : change of variable
- Permalink:
- http://dlmf.nist.gov/22.9.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22
22.9.15
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- Symbols:
- : modulus, : cyclic quantity and : change of variable
- Permalink:
- http://dlmf.nist.gov/22.9.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22
13: 22.2 Definitions
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►With
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22.2.3
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : complex, : modulus and : change of variable
- Permalink:
- http://dlmf.nist.gov/22.2.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.2 and Ch.22
22.2.4
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ⓘ
- Defines:
- : Jacobian elliptic function and : Jacobian elliptic function
- Symbols:
- : theta function, : nome, : complex, : modulus and : change of variable
- Referenced by:
- §23.6(ii)
- Permalink:
- http://dlmf.nist.gov/22.2.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.2 and Ch.22
22.2.5
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- Defines:
- : Jacobian elliptic function and : Jacobian elliptic function
- Symbols:
- : theta function, : nome, : complex, : modulus and : change of variable
- Permalink:
- http://dlmf.nist.gov/22.2.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.2 and Ch.22
22.2.6
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- Defines:
- : Jacobian elliptic function and : Jacobian elliptic function
- Symbols:
- : theta function, : nome, : complex, : modulus and : change of variable
- Referenced by:
- §22.20(ii)
- Permalink:
- http://dlmf.nist.gov/22.2.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.2 and Ch.22
14: 9.5 Integral Representations
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9.5.6
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- Symbols:
- : Airy function, : the ratio of the circumference of a circle to its diameter, : differential of , : exponential function, : integral, : phase and : complex variable
- Source:
- Reid (1995, (5.4), p. 170, with change of variable and using analytic continuation to complex )
- Referenced by:
- Erratum (V1.0.15) for Equation (9.5.6)
- Permalink:
- http://dlmf.nist.gov/9.5.E6
- Encodings:
- pMML, png, TeX
- Clarification (effective with 1.0.15):
- The validity constraint was added.
- See also:
- Annotations for §9.5(ii), §9.5 and Ch.9
9.5.7
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ⓘ
- Symbols:
- : Airy function, : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : exponential function, : base of natural logarithm, : integral, : phase, : complex variable and : change of variable
- Source:
- Copson (1963, (2.1))
- Referenced by:
- §9.5(ii)
- Permalink:
- http://dlmf.nist.gov/9.5.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.5(ii), §9.5 and Ch.9
9.5.8
.
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- Symbols:
- : Airy function, : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral, : phase, : complex variable and : change of variable
- Proof sketch:
- Follows from (9.6.21) and (13.4.4).
- Referenced by:
- §9.17(iii), §9.5(ii)
- Permalink:
- http://dlmf.nist.gov/9.5.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.5(ii), §9.5 and Ch.9
15: 15.12 Asymptotic Approximations
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►where
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15.12.6
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- Symbols:
- : inverse hyperbolic cosine function, : complex variable and : change of variable
- Permalink:
- http://dlmf.nist.gov/15.12.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.12(iii), §15.12 and Ch.15
15.12.10
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ⓘ
- Symbols:
- : inverse hyperbolic cosine function, : complex variable and : change of variable
- Permalink:
- http://dlmf.nist.gov/15.12.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.12(iii), §15.12 and Ch.15
15.12.11
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- Symbols:
- : base of natural logarithm, : principal branch of logarithm function, : change of variable and
- Permalink:
- http://dlmf.nist.gov/15.12.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.12(iii), §15.12 and Ch.15
15.12.13
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- Symbols:
- : base of natural logarithm, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, : change of variable, and : function
- Permalink:
- http://dlmf.nist.gov/15.12.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §15.12(iii), §15.12 and Ch.15
16: 8.18 Asymptotic Expansions of
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►Let
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8.18.2
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- Symbols:
- : principal branch of logarithm function, : variable and : change of variable
- Permalink:
- http://dlmf.nist.gov/8.18.E2
- Encodings:
- pMML, png, TeX
- 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
17: 9.8 Modulus and Phase
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►In terms of Bessel functions, and with ,
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9.8.9
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- Symbols:
- : Bessel function of the first kind, : Bessel function of the second kind, : Airy modulus function, : real variable and : change of variable
- Proof sketch:
- Combine definition (9.8.3) with (9.6.17) and (9.6.19) using (10.4.3).
- Referenced by:
- (9.8.20), (9.8.21), (9.8.22), (9.8.23)
- Permalink:
- http://dlmf.nist.gov/9.8.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.8(i), §9.8 and Ch.9
9.8.10
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- Symbols:
- : Bessel function of the first kind, : Bessel function of the second kind, : Airy modulus function, : real variable and : change of variable
- Proof sketch:
- Combine definition (9.8.7) with (9.6.18) and (9.6.20) using (10.4.3).
- Permalink:
- http://dlmf.nist.gov/9.8.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.8(i), §9.8 and Ch.9
9.8.11
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- Symbols:
- : Bessel function of the first kind, : Bessel function of the second kind, : the ratio of the circumference of a circle to its diameter, : arctangent function, : Airy phase function, : real variable and : change of variable
- Proof sketch:
- Combine definition (9.8.4) with (9.6.17) and (9.6.19) using (10.4.3).
- Permalink:
- http://dlmf.nist.gov/9.8.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.8(i), §9.8 and Ch.9
9.8.12
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- Symbols:
- : Bessel function of the first kind, : Bessel function of the second kind, : the ratio of the circumference of a circle to its diameter, : arctangent function, : Airy phase function, : real variable and : change of variable
- Proof sketch:
- Combine definition (9.8.8) with (9.6.18) and (9.6.20) using (10.4.3).
- Referenced by:
- (9.8.20), (9.8.21), (9.8.22), (9.8.23)
- Permalink:
- http://dlmf.nist.gov/9.8.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §9.8(i), §9.8 and Ch.9
18: 7.7 Integral Representations
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7.7.13
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- Symbols:
- : gamma function, : auxiliary function for Fresnel integrals, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : complex variable, : change of variable and : constant
- Keywords:
- Mellin transform
- Referenced by:
- §7.7(ii), §7.7(ii)
- Permalink:
- http://dlmf.nist.gov/7.7.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7
7.7.14
►In (7.7.13) and (7.7.14) the integration paths are straight lines, , and is a constant such that in (7.7.13), and in (7.7.14).
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- Symbols:
- : gamma function, : auxiliary function for Fresnel integrals, : the ratio of the circumference of a circle to its diameter, : differential of , : imaginary unit, : integral, : complex variable, : change of variable and : constant
- Keywords:
- Mellin transform
- Referenced by:
- §7.7(ii), §7.7(ii)
- Permalink:
- http://dlmf.nist.gov/7.7.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7
19: 29.15 Fourier Series and Chebyshev Series
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29.15.1
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- Symbols:
- : Lamé polynomial, : cosine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, : change of variable and : coefficients
- Referenced by:
- §29.15(i), §29.15(ii)
- Permalink:
- http://dlmf.nist.gov/29.15.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29
29.15.8
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- Symbols:
- : Lamé polynomial, : cosine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, : change of variable and : coefficients
- Referenced by:
- §29.15(i)
- Permalink:
- http://dlmf.nist.gov/29.15.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29
29.15.13
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- Symbols:
- : Lamé polynomial, : sine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, : change of variable and : coefficents
- Referenced by:
- §29.15(i)
- Permalink:
- http://dlmf.nist.gov/29.15.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29
29.15.18
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- Symbols:
- : Jacobian elliptic function, : Lamé polynomial, : cosine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, : change of variable and : coefficents
- Referenced by:
- §29.15(i)
- Permalink:
- http://dlmf.nist.gov/29.15.E18
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29
29.15.23
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- Symbols:
- : Lamé polynomial, : sine function, : nonnegative integer, : nonnegative integer, : nonnegative integer, : complex variable, : real parameter, : change of variable and : coefficents
- Referenced by:
- §29.15(i)
- Permalink:
- http://dlmf.nist.gov/29.15.E23
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29
20: 12.11 Zeros
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12.11.5
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- Symbols:
- : change of variable and : coefficients
- Permalink:
- http://dlmf.nist.gov/12.11.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §12.11(iii), §12.11 and Ch.12
12.11.6
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- Symbols:
- : change of variable and : coefficients
- Permalink:
- http://dlmf.nist.gov/12.11.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §12.11(iii), §12.11 and Ch.12
12.11.8
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- Symbols:
- : change of variable
- Permalink:
- http://dlmf.nist.gov/12.11.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §12.11(iii), §12.11 and Ch.12