complex modulus
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1—10 of 391 matching pages
1: 19.11 Addition Theorems
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19.11.1
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ⓘ
- Symbols:
- : Legendre’s incomplete elliptic integral of the first kind, : real or complex argument, : real or complex modulus, : amplitude and : angle
- Referenced by:
- §19.11(i)
- Permalink:
- http://dlmf.nist.gov/19.11.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.11(i), §19.11 and Ch.19
19.11.2
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ⓘ
- Symbols:
- : Legendre’s incomplete elliptic integral of the second kind, : sine function, : real or complex argument, : real or complex modulus, : amplitude and : angle
- Permalink:
- http://dlmf.nist.gov/19.11.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.11(i), §19.11 and Ch.19
19.11.7
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ⓘ
- Symbols:
- : Legendre’s complete elliptic integral of the first kind, : Legendre’s incomplete elliptic integral of the first kind, : real or complex argument, : real or complex modulus and : amplitude
- Permalink:
- http://dlmf.nist.gov/19.11.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.11(ii), §19.11 and Ch.19
19.11.12
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ⓘ
- Symbols:
- : Legendre’s incomplete elliptic integral of the first kind, : real or complex modulus, : amplitude and : angle
- Permalink:
- http://dlmf.nist.gov/19.11.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.11(iii), §19.11 and Ch.19
19.11.14
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- Defines:
- : angle (locally)
- Symbols:
- : sine function, : real or complex modulus, : amplitude and
- Permalink:
- http://dlmf.nist.gov/19.11.E14
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.11(iii), §19.11 and Ch.19
2: 19.4 Derivatives and Differential Equations
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19.4.3
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ⓘ
- Symbols:
- : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : derivative of with respect to , : real or complex modulus and : complementary modulus
- Permalink:
- http://dlmf.nist.gov/19.4.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.4(i), §19.4 and Ch.19
19.4.5
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ⓘ
- Symbols:
- : cosine function, : Legendre’s incomplete elliptic integral of the first kind, : Legendre’s incomplete elliptic integral of the second kind, : partial derivative of with respect to , : partial differential of , : sine function, : real or complex argument, : real or complex modulus and : complementary modulus
- Permalink:
- http://dlmf.nist.gov/19.4.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.4(i), §19.4 and Ch.19
19.4.6
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ⓘ
- Symbols:
- : Legendre’s incomplete elliptic integral of the first kind, : Legendre’s incomplete elliptic integral of the second kind, : partial derivative of with respect to , : partial differential of , : real or complex argument and : real or complex modulus
- Referenced by:
- §19.30(i)
- Permalink:
- http://dlmf.nist.gov/19.4.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.4(i), §19.4 and Ch.19
19.4.8
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ⓘ
- Symbols:
- : cosine function, : Legendre’s incomplete elliptic integral of the first kind, : sine function, : real or complex argument, : real or complex modulus, : complementary modulus and : differential operator
- Referenced by:
- §19.4(ii)
- Permalink:
- http://dlmf.nist.gov/19.4.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.4(ii), §19.4 and Ch.19
19.4.9
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ⓘ
- Symbols:
- : cosine function, : Legendre’s incomplete elliptic integral of the second kind, : sine function, : real or complex argument, : real or complex modulus, : complementary modulus and : differential operator
- Permalink:
- http://dlmf.nist.gov/19.4.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.4(ii), §19.4 and Ch.19
3: 20.12 Mathematical Applications
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►The space of complex tori (that is, the set of complex numbers in which two of these numbers and are regarded as equivalent if there exist integers such that ) is mapped into the projective space via the identification .
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4: 19.9 Inequalities
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19.9.2
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ⓘ
- Symbols:
- : Legendre’s complete elliptic integral of the first kind, : principal branch of logarithm function, : real or complex modulus and : complementary modulus
- Referenced by:
- §19.9(i), §19.9(i)
- Permalink:
- http://dlmf.nist.gov/19.9.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.9(i), §19.9 and Ch.19
19.9.9
, .
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ⓘ
- Symbols:
- : Legendre’s complete elliptic integral of the second kind, : real or complex modulus and : perimeter
- Referenced by:
- §19.9(i)
- Permalink:
- http://dlmf.nist.gov/19.9.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.9(i), §19.9 and Ch.19
19.9.11
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ⓘ
- Symbols:
- : Legendre’s incomplete elliptic integral of the first kind, : inverse Gudermannian function, : real or complex argument, : real or complex modulus and
- Permalink:
- http://dlmf.nist.gov/19.9.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.9(ii), §19.9 and Ch.19
19.9.12
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ⓘ
- Symbols:
- : Legendre’s incomplete elliptic integral of the second kind, : sine function, : real or complex argument, : real or complex modulus and
- Permalink:
- http://dlmf.nist.gov/19.9.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.9(ii), §19.9 and Ch.19
19.9.17
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ⓘ
- Symbols:
- : Legendre’s incomplete elliptic integral of the first kind, : real or complex argument, : real or complex modulus, : lower limit and : upper limit
- Permalink:
- http://dlmf.nist.gov/19.9.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.9(ii), §19.9 and Ch.19
5: 19.5 Maclaurin and Related Expansions
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19.5.5
, .
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : Legendre’s complementary complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the first kind, : exponential function, : nome and : real or complex modulus
- Referenced by:
- §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.6
,
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ⓘ
- Symbols:
- : nome and : real or complex modulus
- Referenced by:
- §19.5, §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.8
,
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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, : nome, : nonnegative integer and : real or complex modulus
- Referenced by:
- §19.5, §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.10
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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, : nonnegative integer and : real or complex modulus
- Referenced by:
- §19.5
- Permalink:
- http://dlmf.nist.gov/19.5.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
19.5.11
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- Symbols:
- : nonnegative integer and : real or complex modulus
- Permalink:
- http://dlmf.nist.gov/19.5.E11
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.5 and Ch.19
6: 22.17 Moduli Outside the Interval [0,1]
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22.17.1
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ⓘ
- Symbols:
- : generic Jacobian elliptic function, : complex and : modulus
- Permalink:
- http://dlmf.nist.gov/22.17.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.17(i), §22.17 and Ch.22
22.17.6
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ⓘ
- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : imaginary unit, : complex, : modulus, : change of variable and : change of variable
- A&S Ref:
- 16.10.2 (modified)
- Permalink:
- http://dlmf.nist.gov/22.17.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.17(i), §22.17 and Ch.22
22.17.7
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ⓘ
- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : imaginary unit, : complex, : modulus, : change of variable and : change of variable
- A&S Ref:
- 16.10.3 (modified)
- Permalink:
- http://dlmf.nist.gov/22.17.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.17(i), §22.17 and Ch.22
22.17.8
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ⓘ
- Symbols:
- : Jacobian elliptic function, : Jacobian elliptic function, : imaginary unit, : complex, : modulus, : change of variable and : change of variable
- A&S Ref:
- 16.10.4 (modified)
- Permalink:
- http://dlmf.nist.gov/22.17.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.17(i), §22.17 and Ch.22
§22.17(ii) Complex Moduli
…7: 19.8 Quadratic Transformations
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19.8.5
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ⓘ
- Symbols:
- : arithmetic-geometric mean, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : real or complex modulus and : complementary modulus
- Permalink:
- http://dlmf.nist.gov/19.8.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.8(i), §19.8 and Ch.19
19.8.6
, , ,
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ⓘ
- Symbols:
- : arithmetic-geometric mean, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the second kind, : nonnegative integer, : real or complex modulus, : complementary modulus, : iterate and : iterate
- Permalink:
- http://dlmf.nist.gov/19.8.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.8(i), §19.8 and Ch.19
19.8.7
, ,
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ⓘ
- Symbols:
- : arithmetic-geometric mean, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the third kind, : nonnegative integer, : real or complex modulus, : complementary modulus, : real or complex parameter and
- Referenced by:
- §19.8(i)
- Permalink:
- http://dlmf.nist.gov/19.8.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.8(i), §19.8 and Ch.19
19.8.9
, ,
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ⓘ
- Symbols:
- : arithmetic-geometric mean, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the third kind, : nonnegative integer, : real or complex modulus, : complementary modulus, : real or complex parameter and
- Referenced by:
- §19.8(i)
- Permalink:
- http://dlmf.nist.gov/19.8.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.8(i), §19.8 and Ch.19
19.8.10
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- Symbols:
- : real or complex modulus and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/19.8.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.8(i), §19.8 and Ch.19
8: 19.25 Relations to Other Functions
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19.25.13
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ⓘ
- Symbols:
- : elliptic integral symmetric in only two variables, : incomplete elliptic integral of Legendre’s type, : real or complex argument and : real or complex modulus
- Referenced by:
- §19.25(i), §19.36(i)
- Permalink:
- http://dlmf.nist.gov/19.25.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.25(i), §19.25 and Ch.19
19.25.17
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ⓘ
- Symbols:
- : symmetric elliptic integral of first kind, : Legendre’s incomplete elliptic integral of the first kind, : real or complex argument and : real or complex modulus
- Referenced by:
- §19.15
- Permalink:
- http://dlmf.nist.gov/19.25.E17
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.25(i), §19.25 and Ch.19
19.25.18
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- Symbols:
- : real or complex modulus
- Permalink:
- http://dlmf.nist.gov/19.25.E18
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.25(i), §19.25 and Ch.19
19.25.20
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- Symbols:
- : Bulirsch’s incomplete elliptic integral of the first kind, : symmetric elliptic integral of first kind, : real or complex modulus and : inverse
- Permalink:
- http://dlmf.nist.gov/19.25.E20
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.25(ii), §19.25 and Ch.19
19.25.24
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- Symbols:
- : symmetric elliptic integral of first kind, : Legendre’s incomplete elliptic integral of the first kind, : real or complex argument and : real or complex modulus
- Referenced by:
- §19.14(i), §19.25(iii)
- Permalink:
- http://dlmf.nist.gov/19.25.E24
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.25(iii), §19.25 and Ch.19
9: 19.6 Special Cases
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19.6.3
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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, : Legendre’s complete elliptic integral of the third kind, : real or complex modulus and : real or complex parameter
- Permalink:
- http://dlmf.nist.gov/19.6.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.6(i), §19.6 and Ch.19
19.6.5
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ⓘ
- Symbols:
- : Legendre’s complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the third kind, : real or complex modulus and : real or complex parameter
- Referenced by:
- §19.12, Figure 19.3.6, Figure 19.3.6, Figure 19.3.6, §19.6(i), §19.7(iii), §19.8(i)
- Permalink:
- http://dlmf.nist.gov/19.6.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.6(i), §19.6 and Ch.19
19.6.10
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- Symbols:
- : Legendre’s incomplete elliptic integral of the second kind, : real or complex argument and : real or complex modulus
- Permalink:
- http://dlmf.nist.gov/19.6.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §19.6(iii), §19.6 and Ch.19