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Cauchy principal values

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1: 4.10 Integrals
4.10.7 0 x d t ln t = li ( x ) , x > 1 .
The left-hand side of (4.10.7) is a Cauchy principal value1.4(v)). …
2: 1.4 Calculus of One Variable
Cauchy Principal Values
1.4.24 a b f ( x ) d x = P a b f ( x ) d x = lim ϵ 0 + ( a c - ϵ f ( x ) d x + c + ϵ b f ( x ) d x ) ,
1.4.25 - f ( x ) d x = P - f ( x ) d x = lim b - b b f ( x ) d x ,
3: 19.3 Graphics
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Figure 19.3.2: R C ( x , 1 ) and the Cauchy principal value of R C ( x , - 1 ) for 0 x 5 . … Magnify
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Figure 19.3.5: Π ( α 2 , k ) as a function of k 2 and α 2 for - 2 k 2 < 1 , - 2 α 2 2 . Cauchy principal values are shown when α 2 > 1 . … Magnify 3D Help
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Figure 19.3.6: Π ( ϕ , 2 , k ) as a function of k 2 and sin 2 ϕ for - 1 k 2 3 , 0 sin 2 ϕ < 1 . Cauchy principal values are shown when sin 2 ϕ > 1 2 . …If sin 2 ϕ = 1 ( > k 2 ), then the function reduces to Π ( 2 , k ) with Cauchy principal value K ( k ) - Π ( 1 2 k 2 , k ) , which tends to - as k 2 1 - . …If sin 2 ϕ = 1 / k 2 ( < 1 ), then by (19.7.4) it reduces to Π ( 2 / k 2 , 1 / k ) / k , k 2 2 , with Cauchy principal value ( K ( 1 / k ) - Π ( 1 2 , 1 / k ) ) / k , 1 < k 2 < 2 , by (19.6.5). … Magnify 3D Help
4: 19.17 Graphics
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Figure 19.17.6: Cauchy principal value of R J ( x , y , 1 , - 0.5 ) for 0 x 1 , y = 0 ,  0.1 ,  0.5 ,  1 . … Magnify
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Figure 19.17.7: Cauchy principal value of R J ( 0.5 , y , 1 , p ) for y = 0 ,  0.01 ,  0.05 ,  0.2 ,  1 , - 1 p < 0 . … Magnify
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Figure 19.17.8: R J ( 0 , y , 1 , p ) , 0 y 1 , - 1 p 2 . Cauchy principal values are shown when p < 0 . … Magnify 3D Help
5: 4.2 Definitions
6: 19.2 Definitions
The integral for E ( ϕ , k ) is well defined if k 2 = sin 2 ϕ = 1 , and the Cauchy principal value1.4(v)) of Π ( ϕ , α 2 , k ) is taken if 1 - α 2 sin 2 ϕ vanishes at an interior point of the integration path. … If - < p < 0 , then the integral in (19.2.11) is a Cauchy principal value. … where the Cauchy principal value is taken if y < 0 . Formulas involving Π ( ϕ , α 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ( x , y ) . … The Cauchy principal value is hyperbolic: …
7: 19.6 Special Cases
If 1 < α 2 < , then the Cauchy principal value satisfies … Circular and hyperbolic cases, including Cauchy principal values, are unified by using R C ( x , y ) . … For the Cauchy principal value of Π ( ϕ , α 2 , k ) when α 2 > c , see §19.7(iii). …
8: 6.2 Definitions and Interrelations
6.2.5 Ei ( x ) = - - x e - t t d t = - x e t t d t ,
6.2.8 li ( x ) = 0 x d t ln t = Ei ( ln x ) , x > 1 .
9: 9.10 Integrals
9.10.19 Bi ( x ) = 3 x 5 / 4 e ( 2 / 3 ) x 3 / 2 2 π 0 t - 3 / 4 e - ( 2 / 3 ) t 3 / 2 Ai ( t ) x 3 / 2 - t 3 / 2 d t , x > 0 ,
where the last integral is a Cauchy principal value1.4(v)). …
10: 9.12 Scorer Functions
9.12.23 Gi ( x ) = 4 x 2 3 3 / 2 π 2 0 K 1 / 3 ( t ) ζ 2 - t 2 d t , x > 0 ,
where the last integral is a Cauchy principal value1.4(v)). …