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principal value (or branch)

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21: 11.2 Definitions

… ►Principal values correspond to principal values of

(\tfrac{1}{2}z)^{\nu+1}

; compare §4.2(i). … ►Principal values of

\mathbf{K}_{\nu}\left(z\right)

and

\mathbf{M}_{\nu}\left(z\right)

correspond to principal values of the functions on the right-hand sides of (11.2.5) and (11.2.6). ►Unless indicated otherwise,

\mathbf{H}_{\nu}\left(z\right)

,

\mathbf{K}_{\nu}\left(z\right)

,

\mathbf{L}_{\nu}\left(z\right)

, and

\mathbf{M}_{\nu}\left(z\right)

assume their principal values throughout the DLMF. …

22: 4.23 Inverse Trigonometric Functions

… ►

§4.23(ii) Principal Values

… ►Compare the principal value of the logarithm (§4.2(i)). … ►The principal values of the inverse cosecant, secant, and cotangent are given by … ►Graphs of the principal values for real arguments are given in §4.15. … ►Throughout this subsection all quantities assume their principal values. …

23: 10.25 Definitions

… ►

§10.25(ii) Standard Solutions

… ►In particular, the principal branch of

I_{\nu}\left(z\right)

is defined in a similar way: it corresponds to the principal value of

(\tfrac{1}{2}z)^{\nu}

, is analytic in

\mathbb{C}\setminus(-\infty,0]

, and two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. … ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in

\mathbb{C}\setminus(-\infty,0]

, and two-valued and discontinuous on the cut

\operatorname{ph}z=\pm\pi

. … ►Except where indicated otherwise it is assumed throughout the DLMF that the symbols

I_{\nu}\left(z\right)

and

K_{\nu}\left(z\right)

denote the principal values of these functions. …

24: 10.3 Graphics

… ►

See accompanying text — Figure 10.3.10: ${H^{(1)}_{0}}\left(x+iy\right)$ , $-10\leq x\leq 5$ , $-2.8\leq y\leq 4$ . Principal value. … Magnify 3D Help

… ►

See accompanying text — Figure 10.3.12: ${H^{(1)}_{1}}\left(x+iy\right)$ , $-10\leq x\leq 5$ , $-2.8\leq y\leq 4$ . Principal value. … Magnify 3D Help

… ►

See accompanying text — Figure 10.3.14: ${H^{(1)}_{5}}\left(x+iy\right)$ , $-20\leq x\leq 10$ , $-4\leq y\leq 4$ . Principal value. … Magnify 3D Help

►

See accompanying text — Figure 10.3.15: $J_{5.5}\left(x+iy\right)$ , $-10\leq x\leq 10$ , $-4\leq y\leq 4$ . Principal value. … Magnify 3D Help

►

See accompanying text — Figure 10.3.16: ${H^{(1)}_{5.5}}\left(x+iy\right)$ , $-20\leq x\leq 10$ , $-4\leq y\leq 4$ . Principal value. … Magnify 3D Help

…

25: 9.12 Scorer Functions

… ►

9.12.23

\operatorname{Gi}\left(x\right)=\frac{4x^{2}}{3^{3/2}\pi^{2}}\pvint_{0}^{% \infty}\frac{K_{1/3}\left(t\right)}{\zeta^{2}-t^{2}}\,\mathrm{d}t,

x>0

,

ⓘ

Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $x$ : real variable and $\zeta$ : variable
Source:: Gordon (1970, Appendix A)
Permalink:: http://dlmf.nist.gov/9.12.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

►where the last integral is a Cauchy principal value (§1.4(v)). …

26: 8.21 Generalized Sine and Cosine Integrals

… ►

§8.21(ii) Definitions: Principal Values

►When

\operatorname{ph}z=0

(and when

a\neq-1,-3,-5,\dots

, in the case of

\operatorname{Si}\left(a,z\right)

, or

a\neq 0,-2,-4,\dots

, in the case of

\operatorname{Ci}\left(a,z\right)

) the principal values of

\operatorname{si}\left(a,z\right)

,

\operatorname{ci}\left(a,z\right)

,

\operatorname{Si}\left(a,z\right)

, and

\operatorname{Ci}\left(a,z\right)

are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). Elsewhere in the sector

|\operatorname{ph}z|\leq\pi

the principal values are defined by analytic continuation from

\operatorname{ph}z=0

; compare §4.2(i). ►From here on it is assumed that unless indicated otherwise the functions

\operatorname{si}\left(a,z\right)

,

\operatorname{ci}\left(a,z\right)

,

\operatorname{Si}\left(a,z\right)

, and

\operatorname{Ci}\left(a,z\right)

have their principal values. …

27: 8.19 Generalized Exponential Integral

… ►When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of

E_{p}\left(z\right)

, and unless indicated otherwise in the DLMF principal values are assumed. … ►

See accompanying text — Figure 8.19.2: $E_{\frac{1}{2}}\left(x+iy\right)$ , $-4\leq x\leq 4$ , $-4\leq y\leq 4$ . Principal value. … Magnify 3D Help

►

See accompanying text — Figure 8.19.3: $E_{1}\left(x+iy\right)$ , $-4\leq x\leq 4$ , $-4\leq y\leq 4$ . Principal value. … Magnify 3D Help

►

See accompanying text — Figure 8.19.4: $E_{\frac{3}{2}}\left(x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. … Magnify 3D Help

►

See accompanying text — Figure 8.19.5: $E_{2}\left(x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. … Magnify 3D Help

…

28: 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)). …

29: 22.15 Inverse Functions

… ►The principal values satisfy …and unless stated otherwise it is assumed that the inverse functions assume their principal values. …

30: 8.3 Graphics

… ►

See accompanying text — Figure 8.3.8: $\Gamma\left(0.25,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. … Magnify 3D Help

►

See accompanying text — Figure 8.3.9: $\gamma\left(0.25,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. … Magnify 3D Help

… ►

See accompanying text — Figure 8.3.14: $\Gamma\left(2.5,x+iy\right)$ , $-2.2\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. … Magnify 3D Help

►

See accompanying text — Figure 8.3.15: $\gamma\left(2.5,x+iy\right)$ , $-2.2\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. … Magnify 3D Help

…

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