Cauchy–Schwarz

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21: 19.20 Special Cases

… ►Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. If

x, y, z

are permuted so that

0\leq x<y<z

, then the Cauchy principal value of

R_{J}

is given by …

22: 4.2 Definitions

… ► …

23: 19.7 Connection Formulas

… ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of

\Pi\left(\phi,\alpha^{2},k\right)

when

\alpha^{2}>{\csc}^{2}\phi

(see (19.6.5) for the complete case). …

24: 19.21 Connection Formulas

… ►The latter case allows evaluation of Cauchy principal values (see (19.20.14)). …

25: 18.17 Integrals

… ►

18.17.42

\pvint_{-1}^{1}T_{n}\left(y\right)\frac{(1-y^{2})^{-\frac{1}{2}}}{y-x}\,% \mathrm{d}y=\pi U_{n-1}\left(x\right),

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.13.3
Referenced by:: §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E42
Encodings:: pMML, png, TeX
See also:: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

►

18.17.43

\pvint_{-1}^{1}U_{n-1}\left(y\right)\frac{(1-y^{2})^{\frac{1}{2}}}{y-x}\,% \mathrm{d}y=-\pi T_{n}\left(x\right).

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $y$ : real variable, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.13.4
Referenced by:: §18.17(viii)
Permalink:: http://dlmf.nist.gov/18.17.E43
Encodings:: pMML, png, TeX
See also:: Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

►These integrals are Cauchy principal values (§1.4(v)). …

26: 3.4 Differentiation

… ►If

f

can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …

27: 19.8 Quadratic Transformations

… ►If

\alpha^{2}>1

, then the Cauchy principal value is …

28: 19.16 Definitions

… ►In (19.16.2) the Cauchy principal value is taken when

p

is real and negative. …

29: 19.22 Quadratic Transformations

… ►If the last variable of

R_{J}

is negative, then the Cauchy principal value is …

30: 19.36 Methods of Computation

… ►All cases of

R_{F}

R_{C}

R_{J}

, and

R_{D}

are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). …