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21: 19.21 Connection Formulas
§19.21 Connection Formulas
§19.21(i) Complete Integrals
The complete case of R J can be expressed in terms of R F and R D : …
§19.21(ii) Incomplete Integrals
§19.21(iii) Change of Parameter of R J
22: 6.19 Tables
§6.19(ii) Real Variables
  • Abramowitz and Stegun (1964, Chapter 5) includes x 1 Si ( x ) , x 2 Cin ( x ) , x 1 Ein ( x ) , x 1 Ein ( x ) , x = 0 ( .01 ) 0.5 ; Si ( x ) , Ci ( x ) , Ei ( x ) , E 1 ( x ) , x = 0.5 ( .01 ) 2 ; Si ( x ) , Ci ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x = 2 ( .1 ) 10 ; x f ( x ) , x 2 g ( x ) , x e x Ei ( x ) , x e x E 1 ( x ) , x 1 = 0 ( .005 ) 0.1 ; Si ( π x ) , Cin ( π x ) , x = 0 ( .1 ) 10 . Accuracy varies but is within the range 8S–11S.

  • Zhang and Jin (1996, pp. 652, 689) includes Si ( x ) , Ci ( x ) , x = 0 ( .5 ) 20 ( 2 ) 30 , 8D; Ei ( x ) , E 1 ( x ) , x = [ 0 , 100 ] , 8S.

  • Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of z e z E 1 ( z ) , x = 19 ( 1 ) 20 , y = 0 ( 1 ) 20 , 6D; e z E 1 ( z ) , x = 4 ( .5 ) 2 , y = 0 ( .2 ) 1 , 6D; E 1 ( z ) + ln z , x = 2 ( .5 ) 2.5 , y = 0 ( .2 ) 1 , 6D.

  • Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of E 1 ( z ) , ± x = 0.5 , 1 , 3 , 5 , 10 , 15 , 20 , 50 , 100 , y = 0 ( .5 ) 1 ( 1 ) 5 ( 5 ) 30 , 50 , 100 , 8S.

  • 23: 7.22 Methods of Computation
    §7.22(i) Main Functions
    The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. …
    §7.22(ii) Goodwin–Staton Integral
    §7.22(iii) Repeated Integrals of the Complementary Error Function
    The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing i n erfc ( x ) . …
    24: 19.15 Advantages of Symmetry
    §19.15 Advantages of Symmetry
    Symmetry in x , y , z of R F ( x , y , z ) , R G ( x , y , z ) , and R J ( x , y , z , p ) replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). … … For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …
    25: 19.1 Special Notation
    l , m , n nonnegative integers.
    All derivatives are denoted by differentials, not by primes. The first set of main functions treated in this chapter are Legendre’s complete integrals …of the first, second, and third kinds, respectively, and Legendre’s incomplete integralsThe first three functions are incomplete integrals of the first, second, and third kinds, and the cel function includes complete integrals of all three kinds.
    26: 36.9 Integral Identities
    §36.9 Integral Identities
    36.9.1 | Ψ 1 ( x ) | 2 = 2 5 / 3 0 Ψ 1 ( 2 2 / 3 ( 3 u 2 + x ) ) d u ;
    36.9.8 | Ψ ( H ) ( x , y , z ) | 2 = 8 π 2 ( 2 9 ) 1 / 3 Ai ( ( 4 3 ) 1 / 3 ( x + z v + 3 u 2 ) ) Ai ( ( 4 3 ) 1 / 3 ( y + z u + 3 v 2 ) ) d u d v .
    For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). …
    27: 6.4 Analytic Continuation
    §6.4 Analytic Continuation
    Analytic continuation of the principal value of E 1 ( z ) yields a multi-valued function with branch points at z = 0 and z = . The general value of E 1 ( z ) is given by … Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions E 1 ( z ) , Ci ( z ) , Chi ( z ) , f ( z ) , and g ( z ) assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.
    28: 36.10 Differential Equations
    §36.10 Differential Equations
    §36.10(i) Equations for Ψ K ( 𝐱 )
    K = 2 , cusp: … K = 3 , swallowtail: … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( 𝐱 ) satisfy the following operator equations …
    29: 6.11 Relations to Other Functions
    Incomplete Gamma Function
    6.11.1 E 1 ( z ) = Γ ( 0 , z ) .
    Confluent Hypergeometric Function
    6.11.2 E 1 ( z ) = e z U ( 1 , 1 , z ) ,
    30: 6.7 Integral Representations
    §6.7 Integral Representations
    §6.7(i) Exponential Integrals
    §6.7(ii) Sine and Cosine Integrals
    §6.7(iii) Auxiliary Functions