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1: 9.8 Modulus and Phase
§9.8 Modulus and Phase
§9.8(i) Definitions
§9.8(ii) Identities
§9.8(iii) Monotonicity
2: 10.18 Modulus and Phase Functions
§10.18 Modulus and Phase Functions
§10.18(i) Definitions
where M ν ( x ) ( > 0 ) , N ν ( x ) ( > 0 ) , θ ν ( x ) , and ϕ ν ( x ) are continuous real functions of ν and x , with the branches of θ ν ( x ) and ϕ ν ( x ) fixed by …
§10.18(ii) Basic Properties
§10.18(iii) Asymptotic Expansions for Large Argument
3: 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 ) . …
4: 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). …
5: About Color Map
In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of π / 2 correponding to the real and imaginary axes, at more immediately recognizable colors. … We therefore use a piecewise linear mapping as illustrated below, that takes phase 0 to red, π / 2 to yellow, π to cyan and 3 π / 2 to blue. …
6: 33.13 Complex Variable and Parameters
7: 15.6 Integral Representations
In (15.6.3) the point 1 / ( z - 1 ) lies outside the integration contour, the contour cuts the real axis between t = - 1 and 0 , at which point ph t = π and ph ( 1 + t ) = 0 . In (15.6.4) the point 1 / z lies outside the integration contour, and at the point where the contour cuts the negative real axis ph t = π and ph ( 1 - t ) = 0 . …
8: 33.11 Asymptotic Expansions for Large ρ
where θ ( η , ρ ) is defined by (33.2.9), and a and b are defined by (33.8.3). …
F ( η , ρ ) = g ( η , ρ ) cos θ + f ( η , ρ ) sin θ ,
G ( η , ρ ) = f ( η , ρ ) cos θ - g ( η , ρ ) sin θ ,
F ( η , ρ ) = g ^ ( η , ρ ) cos θ + f ^ ( η , ρ ) sin θ ,
Here f 0 = 1 , g 0 = 0 , f ^ 0 = 0 , g ^ 0 = 1 - ( η / ρ ) , and for k = 0 , 1 , 2 , , …
9: 10.27 Connection Formulas
10.27.9 π i J ν ( z ) = e - ν π i / 2 K ν ( z e - π i / 2 ) - e ν π i / 2 K ν ( z e π i / 2 ) , | ph z | 1 2 π .
10.27.10 - π Y ν ( z ) = e - ν π i / 2 K ν ( z e - π i / 2 ) + e ν π i / 2 K ν ( z e π i / 2 ) , | ph z | 1 2 π .
10.27.11 Y ν ( z ) = e ± ( ν + 1 ) π i / 2 I ν ( z e π i / 2 ) - ( 2 / π ) e ν π i / 2 K ν ( z e π i / 2 ) , - 1 2 π ± ph z π .
10: 9.3 Graphics
§9.3(i) Real Variable