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1: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
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 .
This is the principal value; compare (6.2.1). …
2: 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)). …
3: 4.2 Definitions
The principal value, or principal branch, is defined by … We regard this as the closed definition of the principal value. … The principal value is … Another example of a principal value is provided by …
4: 1.4 Calculus of One Variable
1.4.17 x n d x = { x n + 1 n + 1 + C , n 1 , ln | x | + C , n = 1 .
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 ,
5: 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 = . … 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.
6: 4.29 Graphics
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Figure 4.29.2: Principal values of arcsinh x and arccosh x . … Magnify
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Figure 4.29.4: Principal values of arctanh x and arccoth x . … Magnify
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Figure 4.29.6: Principal values of arccsch x and arcsech x . … Magnify
7: 18.40 Methods of Computation
18.40.6 lim ε 0 + a b w ( x ) d x x + i ε x d x = a b w ( x ) d x x x i π w ( x ) ,
8: 4.15 Graphics
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Figure 4.15.2: Arcsin x and Arccos x . Principal values are shown with thickened lines. Magnify
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Figure 4.15.4: arctan x and arccot x . Only principal values are shown. … Magnify
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Figure 4.15.6: arccsc x and arcsec x . Only principal values are shown. … Magnify
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Figure 4.15.9: arcsin ( x + i y ) (principal value). … Magnify 3D Help
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Figure 4.15.11: arctan ( x + i y ) (principal value). … Magnify 3D Help
9: 4.37 Inverse Hyperbolic Functions
§4.37(ii) Principal Values
Compare the principal value of the logarithm (§4.2(i)). … The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by … Graphs of the principal values for real arguments are given in §4.29. … Throughout this subsection all quantities assume their principal values. …
10: 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)). …