# 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 $\pvint^{\infty}_{-\infty}f(x)\,\mathrm{d}x=P\int^{\infty}_{-\infty}f(x)\,% \mathrm{d}x=\lim_{b\to\infty}\int^{b}_{-b}f(x)\,\mathrm{d}x,$
##### 3: 19.3 Graphics Figure 19.3.2: R C ⁡ ( x , 1 ) and the Cauchy principal value of R C ⁡ ( x , − 1 ) for 0 ≤ x ≤ 5 . … Magnify 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 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 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 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 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
##### 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 $f(x)=-\frac{1}{\pi}\pvint^{\infty}_{-\infty}\frac{\mathcal{H}\mskip-3.0muf% \mskip 3.0mu\left(u\right)}{u-x}\,\mathrm{d}u.$