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1: 6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

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6.2.5 $$\mathrm{Ei}\left(x\right)=-{⨍}_{-x}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t}}{t}dt={⨍}_{-\mathrm{\infty }}^x\frac{{\mathrm{e}}^t}{t}dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

5.1.2

Permalink:

http://dlmf.nist.gov/6.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

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6.2.8 $$\mathrm{li}\left(x\right)={⨍}_0^x\frac{dt}{\ln t}=\mathrm{Ei}\left(\ln x\right),$$ $x>1$.

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Defines:

$\mathrm{li}\left(x\right)$: logarithmic integral

Symbols:

$dx$: differential of $x$ , $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

5.1.3

Referenced by:

§27.12, §6.12(i)

Permalink:

http://dlmf.nist.gov/6.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.2(i), §6.2 and Ch.6

… ►This is the principal value; compare (6.2.1). …

2: 4.10 Integrals

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4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

►The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(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

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1.4.17

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $C$: constant and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(iv), §1.4(iv), §1.4 and Ch.1

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Cauchy Principal Values

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1.4.24 $${⨍}_a^{b}f(x)dx=𝑃{\int }_a^{b}f(x)dx=\underset{ϵ\to 0+}{lim}\left({\int }_a^{c-ϵ}f(x)dx+{\int }_{c+ϵ}^{b}f(x)dx\right),$$

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Defines:

${⨍}_a^b$: Cauchy principal value

Symbols:

$dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.18(vi)

Permalink:

http://dlmf.nist.gov/1.4.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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1.4.25 $${⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=𝑃{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=\underset{b\to \mathrm{\infty }}{lim}{\int }_{-b}^{b}f(x)dx,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral and ${⨍}_a^b$: Cauchy principal value

Referenced by:

§1.14(i)

Permalink:

http://dlmf.nist.gov/1.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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5: 9.10 Integrals

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt,$$ $x>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

Proof sketch:

To prove this equation, combine (9.2.10) with (9.10.18).

Referenced by:

§9.5(i), Erratum (V1.0.10) for Equation (9.10.19)

Permalink:

http://dlmf.nist.gov/9.10.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Bi}\left(x\right)=\frac{x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2^{5/2}\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

►where the last integral is a Cauchy principal value (§1.4(v)). …

6: 6.4 Analytic Continuation

… ►Analytic continuation of the principal value of $E_1\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_1\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.

7: 4.29 Graphics

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Figure 4.29.2: Principal values of $\mathrm{arcsinh}x$ and $\mathrm{arccosh}x$. … Magnify

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Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$. … Magnify

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. … Magnify

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8: 18.40 Methods of Computation

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18.40.6 $$\underset{\epsilon \to 0+}{lim}{\int }_a^b\frac{w(x)dx}{x^{\prime }+\mathrm{i}\epsilon -x}dx={⨍}_a^b\frac{w(x)dx}{x^{\prime }-x}-\mathrm{i}\pi w(x^{\prime }),$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${⨍}_a^b$: Cauchy principal value, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

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9: 4.15 Graphics

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Figure 4.15.2: $\mathrm{Arcsin}x$ and $\mathrm{Arccos}x$. Principal values are shown with thickened lines. Magnify

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Figure 4.15.4: $\arctan x$ and $\mathrm{arccot}x$. Only principal values are shown. … Magnify

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Figure 4.15.6: $\mathrm{arccsc}x$ and $\mathrm{arcsec}x$. Only principal values are shown. … Magnify

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Figure 4.15.9: $\arcsin \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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Figure 4.15.11: $\arctan \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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10: 4.37 Inverse Hyperbolic Functions

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§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. …