# Cauchy principal values

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##### 1: 4.10 Integrals
The left-hand side of (4.10.7) is a Cauchy principal value1.4(v)). …
##### 2: 1.4 Calculus of One Variable
###### CauchyPrincipalValues
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 $\pvint^{\infty}_{-\infty}f(x)\,\mathrm{d}x=P\int^{\infty}_{-\infty}f(x)\,% \mathrm{d}x=\lim_{b\to\infty}\int^{b}_{-b}f(x)\,\mathrm{d}x,$
##### 6: 19.2 Definitions
The integral for $E\left(\phi,k\right)$ is well defined if $k^{2}={\sin}^{2}\phi=1$, and the Cauchy principal value1.4(v)) of $\Pi\left(\phi,\alpha^{2},k\right)$ is taken if $1-\alpha^{2}{\sin}^{2}\phi$ vanishes at an interior point of the integration path. … If $-\infty, then the integral in (19.2.11) is a Cauchy principal value. … where the Cauchy principal value is taken if $y<0$. Formulas involving $\Pi\left(\phi,\alpha^{2},k\right)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_{C}\left(x,y\right)$. … The Cauchy principal value is hyperbolic: …
##### 7: 19.6 Special Cases
If $1<\alpha^{2}<\infty$, then the Cauchy principal value satisfies … Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_{C}\left(x,y\right)$. … For the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>c$, see §19.7(iii). …
##### 8: 6.2 Definitions and Interrelations
6.2.8 $\operatorname{li}\left(x\right)=\pvint_{0}^{x}\frac{\,\mathrm{d}t}{\ln t}=% \operatorname{Ei}\left(\ln x\right),$ $x>1$.
##### 9: 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)). …
##### 10: 1.14 Integral Transforms
1.14.3 $\tfrac{1}{2}(f(u+)+f(u-))=\frac{1}{\sqrt{2\pi}}\pvint^{\infty}_{-\infty}F(x)e^% {-ixu}\,\mathrm{d}x,$
where the last integral denotes the Cauchy principal value (1.4.25). …
1.14.41 $\mathcal{H}\left(f\right)\left(x\right)=\mathcal{H}\mskip-3.0muf\mskip 3.0mu% \left(x\right)=\frac{1}{\pi}\pvint^{\infty}_{-\infty}\frac{f(t)}{t-x}\,\mathrm% {d}t,$
1.14.44 $f(x)=-\frac{1}{\pi}\pvint^{\infty}_{-\infty}\frac{\mathcal{H}\mskip-3.0muf% \mskip 3.0mu\left(u\right)}{u-x}\,\mathrm{d}u.$