# principal value

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##### 1: 6.2 Definitions and Interrelations
###### §6.2(i) Exponential and Logarithmic Integrals
6.2.5 $\operatorname{Ei}\left(x\right)=-\,\pvint_{-x}^{\infty}\frac{e^{-t}}{t}\,% \mathrm{d}t=\pvint_{-\infty}^{x}\frac{e^{t}}{t}\,\mathrm{d}t,$
6.2.8 $\operatorname{li}\left(x\right)=\pvint_{0}^{x}\frac{\,\mathrm{d}t}{\ln t}=% \operatorname{Ei}\left(\ln x\right),$ $x>1$.
This is the principal value; compare (6.2.1). …
##### 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 $\int x^{n}\,\mathrm{d}x=\begin{cases}\dfrac{x^{n+1}}{n+1}+C,&\quad n\not=-1,\\ \ln\left|x\right|+C,&\quad n=-1.\end{cases}$
###### Cauchy PrincipalValues
1.4.24 $\pvint^{b}_{a}f(x)\,\mathrm{d}x=P\int^{b}_{a}f(x)\,\mathrm{d}x=\lim_{\epsilon% \to 0+}\left(\int^{c-\epsilon}_{a}f(x)\,\mathrm{d}x+\int^{b}_{c+\epsilon}f(x)% \,\mathrm{d}x\right),$
1.4.25 $\pvint^{\infty}_{-\infty}f(x)\,\mathrm{d}x=P\int^{\infty}_{-\infty}f(x)\,% \mathrm{d}x=\lim_{b\to\infty}\int^{b}_{-b}f(x)\,\mathrm{d}x,$
##### 5: 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 $z=\infty$. … Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_{1}\left(z\right)$, $\operatorname{Ci}\left(z\right)$, $\operatorname{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, 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: 18.40 Methods of Computation
18.40.6 $\lim_{\varepsilon\to 0{+}}\int_{a}^{b}\frac{w(x)\,\mathrm{d}x}{x^{\prime}+% \mathrm{i}\varepsilon-x}\,\mathrm{d}x=\pvint_{a}^{b}\frac{w(x)\,\mathrm{d}x}{x% ^{\prime}-x}-\mathrm{i}\pi w(x^{\prime}),$
##### 9: 4.37 Inverse Hyperbolic Functions
###### §4.37(ii) PrincipalValues
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 $\operatorname{Bi}\left(x\right)=\frac{3x^{5/4}e^{(2/3)x^{3/2}}}{2\pi}\*\pvint_% {0}^{\infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\operatorname{Ai}\left(t\right)}{x^% {3/2}-t^{3/2}}\,\mathrm{d}t,$ $x>0$,
where the last integral is a Cauchy principal value1.4(v)). …