Cauchy

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1: 1.7 Inequalities

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Cauchy–Schwarz Inequality

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Cauchy–Schwarz Inequality

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2: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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Cauchy’s Sum

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3: 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)). …

4: 1.3 Determinants, Linear Operators, and Spectral Expansions

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Cauchy Determinant

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5: 19.3 Graphics

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Figure 19.3.2: $R_C(x,1)$ and the Cauchy principal value of $R_C(x,-1)$ for $0\le x\le 5$. … Magnify

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Figure 19.3.5: $\mathrm{\Pi }({\alpha }^2,k)$ as a function of $k^2$ and ${\alpha }^2$ for , $-2\le {\alpha }^2\le 2$. Cauchy principal values are shown when ${\alpha }^2>1$. … Magnify 3D Help

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . Cauchy principal values are shown when ${\sin }^2\varphi >\frac{1}{2}$. …If ${\sin }^2\varphi =1$ ($>k^2$), then the function reduces to $\mathrm{\Pi }(2,k)$ with Cauchy principal value $K\left(k\right)-\mathrm{\Pi }(\frac{1}{2}{k}^2,k)$, which tends to $-\mathrm{\infty }$ as $k^2\to 1-$. …If ${\sin }^2\varphi =1/k^2$ (), then by (19.7.4) it reduces to $\mathrm{\Pi }(2/k^2,1/k)/k$, $k^2\ne 2$, with Cauchy principal value $(K\left(1/k\right)-\mathrm{\Pi }(\frac{1}{2},1/k))/k$, , by (19.6.5). … Magnify 3D Help

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6: 1.9 Calculus of a Complex Variable

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Cauchy–Riemann Equations

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Cauchy’s Theorem

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Cauchy’s Integral Formula

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7: 1.4 Calculus of One Variable

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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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8: 19.17 Graphics

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Figure 19.17.6: Cauchy principal value of $R_J(x,y,1,-0.5)$ for $0\le x\le 1$, $y=0,\mathrm{\hspace{0.17em}0.1},\mathrm{\hspace{0.17em}0.5},\mathrm{\hspace{0.17em}1}$. … Magnify

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Figure 19.17.7: Cauchy principal value of $R_J(0.5,y,1,p)$ for $y=0,\mathrm{\hspace{0.17em}0.01},\mathrm{\hspace{0.17em}0.05},\mathrm{\hspace{0.17em}0.2},\mathrm{\hspace{0.17em}1}$, . … Magnify

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Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. Cauchy principal values are shown when . … Magnify 3D Help

9: 19.2 Definitions

… ►The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. … ►If , then the integral in (19.2.11) is a Cauchy principal value. … ►where the Cauchy principal value is taken if . Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$. … ►The Cauchy principal value is hyperbolic: …

10: 19.6 Special Cases

… ►If , then the Cauchy principal value satisfies … ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_C(x,y)$. … ►For the Cauchy principal value of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when ${\alpha }^2>c$, see §19.7(iii). …