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reduction to basic elliptic integrals

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1: 19.16 Definitions
§19.16(i) Symmetric Integrals
which is homogeneous and of degree a in the z ’s, and unchanged when the same permutation is applied to both sets of subscripts 1 , , n . …The R -function is often used to make a unified statement of a property of several elliptic integrals. … … Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: …
2: 1.14 Integral Transforms
§1.14 Integral Transforms
If f ( t ) is absolutely integrable on ( , ) , then F ( x ) is continuous, F ( x ) 0 as x ± , and … If also lim t 0 + f ( t ) / t exists, then … Note: If f ( x ) is continuous and α and β are real numbers such that f ( x ) = O ( x α ) as x 0 + and f ( x ) = O ( x β ) as x , then x σ 1 f ( x ) is integrable on ( 0 , ) for all σ ( α , β ) . … Sufficient conditions for the integral to converge are that s is a positive real number, and f ( t ) = O ( t δ ) as t , where δ > 0 . …
3: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity
4: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
§8.19(i) Definition and Integral Representations
In Figures 8.19.28.19.5, height corresponds to the absolute value of the function and color to the phase. …
§8.19(vi) Relation to Confluent Hypergeometric Function
§8.19(x) Integrals
5: 22.15 Inverse Functions
§22.15 Inverse Functions
The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). … Equations (22.15.1) and (22.15.4), for arcsn ( x , k ) , are equivalent to (22.15.12) and also to
§22.15(ii) Representations as Elliptic Integrals
6: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
The logarithmic integral is defined by …
§6.2(ii) Sine and Cosine Integrals
7: 36.2 Catastrophes and Canonical Integrals
Normal Forms for Umbilic Catastrophes with Codimension K = 3
Canonical Integrals
Ψ 1 is related to the Airy function (§9.2): … …
§36.2(iv) Addendum to 36.2(ii) Special Cases
8: 19.2 Definitions
§19.2(i) General Elliptic Integrals
is called an elliptic integral. …
§19.2(ii) Legendre’s Integrals
§19.2(iii) Bulirsch’s Integrals
§19.2(iv) A Related Function: R C ( x , y )
9: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
§8.21(iii) Integral Representations
§8.21(v) Special Values
When z with | ph z | π δ ( < π ), …
10: 22.2 Definitions
§22.2 Definitions
where K ( k ) , K ( k ) are defined in §19.2(ii). … s s ( z , k ) = 1 . … and on the left-hand side of (22.2.11) p , q are any pair of the letters s , c , d , n , and on the right-hand side they correspond to the integers 1 , 2 , 3 , 4 .