About the Project

modulus and phase functions

AdvancedHelp

(0.007 seconds)

1—10 of 134 matching pages

1: 9.8 Modulus and Phase
§9.8(i) Definitions
§9.8(ii) Identities
§9.8(iii) Monotonicity
§9.8(iv) Asymptotic Expansions
2: 10.68 Modulus and Phase Functions
§10.68 Modulus and Phase Functions
§10.68(i) Definitions
§10.68(ii) Basic Properties
§10.68(iii) Asymptotic Expansions for Large Argument
Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …
3: 10.18 Modulus and Phase Functions
§10.18 Modulus and Phase Functions
§10.18(i) Definitions
§10.18(ii) Basic Properties
§10.18(iii) Asymptotic Expansions for Large Argument
In (10.18.17) and (10.18.18) the remainder after n terms does not exceed the ( n + 1 ) th term in absolute value and is of the same sign, provided that n > ν 1 2 for (10.18.17) and 3 2 ν 3 2 for (10.18.18).
4: 12.19 Tables
  • Miller (1955) includes W ( a , x ) , W ( a , x ) , and reduced derivatives for a = 10 ( 1 ) 10 , x = 0 ( .1 ) 10 , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.

  • Fox (1960) includes modulus and phase functions for W ( a , x ) and W ( a , x ) , and several auxiliary functions for x 1 = 0 ( .005 ) 0.1 , a = 10 ( 1 ) 10 , 8S.

  • 5: 10.3 Graphics
    §10.3(i) Real Order and Variable
    For the modulus and phase functions M ν ( x ) , θ ν ( x ) , N ν ( x ) , and ϕ ν ( x ) see §10.18. …
    See accompanying text
    Figure 10.3.4: θ 5 ( x ) , ϕ 5 ( x ) , 0 x 15 . Magnify
    6: 33.13 Complex Variable and Parameters
    33.13.1 C ( η ) = 2 e i σ ( η ) ( π η / 2 ) Γ ( + 1 i η ) / Γ ( 2 + 2 ) ,
    7: 36.3 Visualizations of Canonical Integrals
    §36.3(i) Canonical Integrals: Modulus
    §36.3(ii) Canonical Integrals: Phase
    In Figure 36.3.13(a) points of confluence of phase contours are zeros of Ψ 2 ( x , y ) ; similarly for other contour plots in this subsection. …
    Figure 36.3.13: Phase of Pearcey integral ph Ψ 2 ( x , y ) .
    Figure 36.3.21: Phase of hyperbolic umbilic canonical integral ph Ψ ( H ) ( x , y , 3 ) . …
    8: 32.11 Asymptotic Approximations for Real Variables
    32.11.24 s = ( exp ( π d 2 ) 1 ) 1 / 2 exp ( i ( 3 2 d 2 ln 2 1 4 π + χ ph Γ ( 1 2 i d 2 ) ) ) .
    9: 9.11 Products
    9.11.19 0 d t Ai 2 ( t ) + Bi 2 ( t ) = 0 t d t Ai 2 ( t ) + Bi 2 ( t ) = π 2 6 .
    10: 12.2 Differential Equations
    §12.2(vi) Solution U ¯ ( a , x ) ; Modulus and Phase Functions
    For properties of the modulus and phase functions, including differential equations, see Miller (1955, pp. 72–73). …