# §10.43 Integrals

## §10.43(i) Indefinite Integrals

Let $\mathscr{Z}_{\nu}\left(z\right)$ be defined as in §10.25(ii). Then

 10.43.1 $\displaystyle\int z^{\nu+1}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=z^{\nu+1}\mathscr{Z}_{\nu+1}\left(z\right),$ $\displaystyle\int z^{-\nu+1}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=z^{-\nu+1}\mathscr{Z}_{\nu-1}\left(z\right).$
 10.43.2 $\int z^{\nu}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z=\pi^{\frac{1}{2}}2^{\nu% -1}\Gamma\left(\nu+\tfrac{1}{2}\right)z\*\left(\mathscr{Z}_{\nu}\left(z\right)% \mathbf{L}_{\nu-1}\left(z\right)-\mathscr{Z}_{\nu-1}\left(z\right)\mathbf{L}_{% \nu}\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$.

For the modified Struve function $\mathbf{L}_{\nu}\left(z\right)$ see §11.2(i).

 10.43.3 $\displaystyle\int e^{\pm z}z^{\nu}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=\frac{e^{\pm z}z^{\nu+1}}{2\nu+1}\left(\mathscr{Z}_{\nu}\left(z% \right)\mp\mathscr{Z}_{\nu+1}\left(z\right)\right),$ $\nu\neq-\tfrac{1}{2}$, $\displaystyle\int e^{\pm z}z^{-\nu}\mathscr{Z}_{\nu}\left(z\right)\mathrm{d}z$ $\displaystyle=\frac{e^{\pm z}z^{-\nu+1}}{1-2\nu}\left(\mathscr{Z}_{\nu}\left(z% \right)\mp\mathscr{Z}_{\nu-1}\left(z\right)\right),$ $\nu\neq\tfrac{1}{2}$. ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter A&S Ref: 11.3.12, 11.3.14, 11.3.15, 11.3.17 (modified) Permalink: http://dlmf.nist.gov/10.43.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 10.43(i), 10.43 and 10

## §10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\infty)$

 10.43.4 $\int_{0}^{x}\frac{I_{0}\left(t\right)-1}{t}\mathrm{d}t=\frac{1}{2}\sum_{k=1}^{% \infty}(-1)^{k-1}\frac{\psi\left(k+1\right)-\psi\left(1\right)}{k!}(\tfrac{1}{% 2}x)^{k}I_{k}\left(x\right)=\frac{2}{x}\sum_{k=0}^{\infty}(-1)^{k}(2k+3)(\psi% \left(k+2\right)-\psi\left(1\right))I_{2k+3}\left(x\right).$
 10.43.5 $\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}\mathrm{d}t=\frac{1}{2}\left(\ln% \left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+\frac{\pi^{2}}{24}-\sum_{k=1}^{% \infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)% \right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}},$

where $\psi=\ifrac{\Gamma'}{\Gamma}$ and $\gamma$ is Euler’s constant (§5.2).

 10.43.6 $\int_{0}^{x}e^{-t}I_{n}\left(t\right)\mathrm{d}t=xe^{-x}(I_{0}\left(x\right)+I% _{1}\left(x\right))+n(e^{-x}I_{0}\left(x\right)-1)+2e^{-x}\sum_{k=1}^{n-1}(n-k% )I_{k}\left(x\right),$ $n=0,1,2,\ldots$.
 10.43.7 $\int_{0}^{x}e^{\pm t}t^{\nu}I_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{\pm x}x^% {\nu+1}}{2\nu+1}(I_{\nu}\left(x\right)\mp I_{\nu+1}\left(x\right)),$ $\Re\nu>-\tfrac{1}{2}$,
 10.43.8 $\int_{0}^{x}e^{\pm t}t^{-\nu}I_{\nu}\left(t\right)\mathrm{d}t=-\frac{e^{\pm x}% x^{-\nu+1}}{2\nu-1}(I_{\nu}\left(x\right)\mp I_{\nu-1}\left(x\right))\mp\frac{% 2^{-\nu+1}}{(2\nu-1)\Gamma\left(\nu\right)},$ $\nu\neq\tfrac{1}{2}$.
 10.43.9 $\int_{0}^{x}e^{\pm t}t^{\nu}K_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{\pm x}x^% {\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)\pm K_{\nu+1}\left(x\right))\mp\frac{2^{% \nu}\Gamma\left(\nu+1\right)}{2\nu+1},$ $\Re\nu>-\tfrac{1}{2}$,
 10.43.10 $\int_{x}^{\infty}e^{t}t^{-\nu}K_{\nu}\left(t\right)\mathrm{d}t=\frac{e^{x}x^{-% \nu+1}}{2\nu-1}(K_{\nu}\left(x\right)+K_{\nu-1}\left(x\right)),$ $\Re\nu>\tfrac{1}{2}$.

## §10.43(iii) Fractional Integrals

The Bickley function $\mathrm{Ki}_{\alpha}\left(x\right)$ is defined by

 10.43.11 $\mathrm{Ki}_{\alpha}\left(x\right)=\frac{1}{\Gamma\left(\alpha\right)}\int_{x}% ^{\infty}(t-x)^{\alpha-1}K_{0}\left(t\right)\mathrm{d}t,$ ⓘ Defines: $\mathrm{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$: Bickley function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind and $x$: real variable A&S Ref: 11.2.11 Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E11 Encodings: TeX, pMML, png See also: Annotations for 10.43(iii), 10.43 and 10

when $\Re\alpha>0$ and $x>0$, and by analytic continuation elsewhere. Equivalently,

 10.43.12 $\mathrm{Ki}_{\alpha}\left(x\right)=\int_{0}^{\infty}\frac{e^{-x\cosh t}}{(% \cosh t)^{\alpha}}\mathrm{d}t,$ $x>0$. ⓘ Symbols: $\mathrm{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$: Bickley function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh\NVar{z}$: hyperbolic cosine function, $\int$: integral and $x$: real variable A&S Ref: 11.2.10 (with other conditions) Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E12 Encodings: TeX, pMML, png See also: Annotations for 10.43(iii), 10.43 and 10

### Properties

 10.43.13 $\mathrm{Ki}_{\alpha}\left(x\right)=\int_{x}^{\infty}\mathrm{Ki}_{\alpha-1}% \left(t\right)\mathrm{d}t,$ ⓘ Symbols: $\mathrm{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$: Bickley function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $x$: real variable A&S Ref: 11.2.8 Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E13 Encodings: TeX, pMML, png See also: Annotations for 10.43(iii), 10.43(iii), 10.43 and 10
 10.43.14 $\mathrm{Ki}_{0}\left(x\right)=K_{0}\left(x\right),$ ⓘ Symbols: $\mathrm{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$: Bickley function, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind and $x$: real variable A&S Ref: 11.2.8 Permalink: http://dlmf.nist.gov/10.43.E14 Encodings: TeX, pMML, png See also: Annotations for 10.43(iii), 10.43(iii), 10.43 and 10
 10.43.15 $\mathrm{Ki}_{-n}\left(x\right)=(-1)^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{% n}}K_{0}\left(x\right),$ $n=1,2,3,\ldots$.
 10.43.16 $\mathrm{Ki}_{\alpha}\left(0\right)=\frac{\sqrt{\pi}\Gamma\left(\frac{1}{2}% \alpha\right)}{2\Gamma\left(\frac{1}{2}\alpha+\frac{1}{2}\right)},$ $\alpha\neq 0,-2,-4,\dots$. ⓘ Symbols: $\mathrm{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$: Bickley function, $\Gamma\left(\NVar{z}\right)$: gamma function and $\pi$: the ratio of the circumference of a circle to its diameter A&S Ref: 11.2.12, 11.2.13 (with less restrictive conditions) Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E16 Encodings: TeX, pMML, png See also: Annotations for 10.43(iii), 10.43(iii), 10.43 and 10
 10.43.17 $\alpha\mathrm{Ki}_{\alpha+1}\left(x\right)+x\mathrm{Ki}_{\alpha}\left(x\right)% +(1-\alpha)\mathrm{Ki}_{\alpha-1}\left(x\right)-x\mathrm{Ki}_{\alpha-2}\left(x% \right)=0.$ ⓘ Symbols: $\mathrm{Ki}_{\NVar{\alpha}}\left(\NVar{x}\right)$: Bickley function and $x$: real variable A&S Ref: 11.2.14 Referenced by: §10.43(iii) Permalink: http://dlmf.nist.gov/10.43.E17 Encodings: TeX, pMML, png See also: Annotations for 10.43(iii), 10.43(iii), 10.43 and 10

For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos (1983c, 1989) and Luke (1962, Chapter 8).

## §10.43(iv) Integrals over the Interval ($0,\infty$)

 10.43.18 $\int_{0}^{\infty}K_{\nu}\left(t\right)\mathrm{d}t=\tfrac{1}{2}\pi\sec(\tfrac{1% }{2}\pi\nu),$ $|\Re\nu|<1$.
 10.43.19 $\int_{0}^{\infty}t^{\mu-1}K_{\nu}\left(t\right)\mathrm{d}t=2^{\mu-2}\Gamma% \left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\Gamma\left(\tfrac{1}{2}\mu+\tfrac% {1}{2}\nu\right),$ $|\Re\nu|<\Re\mu$.
 10.43.20 $\displaystyle\int_{0}^{\infty}\cos\left(at\right)K_{0}\left(t\right)\mathrm{d}t$ $\displaystyle=\frac{\pi}{2(1+a^{2})^{\frac{1}{2}}},$ $|\Im a|<1$, 10.43.21 $\displaystyle\int_{0}^{\infty}\sin\left(at\right)K_{0}\left(t\right)\mathrm{d}t$ $\displaystyle=\frac{\operatorname{arcsinh}a}{(1+a^{2})^{\frac{1}{2}}},$ $|\Im a|<1$.

When $\Re\mu>|\Re\nu|$,

 10.43.22 $\int_{0}^{\infty}t^{\mu-1}e^{-at}K_{\nu}\left(t\right)\mathrm{d}t=\begin{cases% }\left(\frac{1}{2}\pi\right)^{\frac{1}{2}}\Gamma\left(\mu-\nu\right)\Gamma% \left(\mu+\nu\right)(1-a^{2})^{-\frac{1}{2}\mu+\frac{1}{4}}\mathsf{P}^{-\mu+% \frac{1}{2}}_{\nu-\frac{1}{2}}\left(a\right),&-1

For the second equation there is a cut in the $a$-plane along the interval $[0,1]$, and all quantities assume their principal values (§4.2(i)). For the Ferrers function $\mathsf{P}$ and the associated Legendre function $P$, see §§14.3(i) and 14.21(i).

 10.43.23 $\displaystyle\int_{0}^{\infty}t^{\nu+1}I_{\nu}\left(bt\right)\exp(-p^{2}t^{2})% \mathrm{d}t$ $\displaystyle=\frac{b^{\nu}}{(2p^{2})^{\nu+1}}\exp\left(\frac{b^{2}}{4p^{2}}% \right),$ $\Re\nu>-1,\Re(p^{2})>0$, 10.43.24 $\displaystyle\int_{0}^{\infty}I_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}% \right)\mathrm{d}t$ $\displaystyle=\frac{\sqrt{\pi}}{2p}\exp\left(\frac{b^{2}}{8p^{2}}\right)I_{% \frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}\right),$ $\Re\nu>-1$, $\Re(p^{2})>0$, 10.43.25 $\displaystyle\int_{0}^{\infty}K_{\nu}\left(bt\right)\exp\left(-p^{2}t^{2}% \right)\mathrm{d}t$ $\displaystyle=\frac{\sqrt{\pi}}{4p}\sec\left(\tfrac{1}{2}\pi\nu\right)\exp% \left(\frac{b^{2}}{8p^{2}}\right)K_{\frac{1}{2}\nu}\left(\frac{b^{2}}{8p^{2}}% \right),$ $|\Re\nu|<1$, $\Re(p^{2})>0$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\exp\NVar{z}$: exponential function, $\int$: integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\Re$: real part, $\sec\NVar{z}$: secant function and $\nu$: complex parameter A&S Ref: 11.4.32 (Case $\nu=0$) Referenced by: §10.43(iv) Permalink: http://dlmf.nist.gov/10.43.E25 Encodings: TeX, pMML, png See also: Annotations for 10.43(iv), 10.43 and 10 10.43.26 $\displaystyle\int_{0}^{\infty}\frac{K_{\mu}\left(at\right)J_{\nu}\left(bt% \right)}{t^{\lambda}}\mathrm{d}t$ $\displaystyle=\frac{b^{\nu}\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda+\frac% {1}{2}\mu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\lambda-% \frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\lambda+1}a^{\nu-\lambda+1}}\*\mathbf{F}% \left(\frac{\nu-\lambda+\mu+1}{2},\frac{\nu-\lambda-\mu+1}{2};\nu+1;-\frac{b^{% 2}}{a^{2}}\right),$ $\Re(\nu+1-\lambda)>|\Re\mu|,\Re a>|\Im b|$.

For the hypergeometric function $\mathbf{F}$ see §15.2(i).

 10.43.27 $\displaystyle\int_{0}^{\infty}t^{\mu+\nu+1}K_{\mu}\left(at\right)J_{\nu}\left(% bt\right)\mathrm{d}t$ $\displaystyle=\frac{(2a)^{\mu}(2b)^{\nu}\Gamma\left(\mu+\nu+1\right)}{(a^{2}+b% ^{2})^{\mu+\nu+1}},$ $\Re(\nu+1)>|\Re\mu|,\Re a>|\Im b|$. 10.43.28 $\displaystyle\int_{0}^{\infty}t\exp(-p^{2}t^{2})I_{\nu}\left(at\right)I_{\nu}% \left(bt\right)\mathrm{d}t$ $\displaystyle=\frac{1}{2p^{2}}\exp\left(\frac{a^{2}+b^{2}}{4p^{2}}\right)I_{% \nu}\left(\frac{ab}{2p^{2}}\right),$ $\Re\nu>-1,\Re(p^{2})>0$, 10.43.29 $\displaystyle\int_{0}^{\infty}t\exp(-p^{2}t^{2})I_{0}\left(at\right)K_{0}\left% (at\right)\mathrm{d}t$ $\displaystyle=\frac{1}{4p^{2}}\exp\left(\frac{a^{2}}{2p^{2}}\right)K_{0}\left(% \frac{a^{2}}{2p^{2}}\right),$ $\Re(p^{2})>0$.

For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b).

## §10.43(v) Kontorovich–Lebedev Transform

The Kontorovich–Lebedev transform of a function $g(x)$ is defined as

 10.43.30 $f(y)=\frac{2y}{\pi^{2}}\sinh(\pi y)\int_{0}^{\infty}\frac{g(x)}{x}K_{iy}\left(% x\right)\mathrm{d}x.$ ⓘ Defines: $f(x)$: Kontorovich–Lebedev transform of $g(x)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\sinh\NVar{z}$: hyperbolic sine function, $\int$: integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $x$: real variable, $y$: real variable and $g(x)$: function Referenced by: §10.43(v) Permalink: http://dlmf.nist.gov/10.43.E30 Encodings: TeX, pMML, png See also: Annotations for 10.43(v), 10.43 and 10

Then

 10.43.31 $g(x)=\int_{0}^{\infty}f(y)K_{iy}\left(x\right)\mathrm{d}y,$

provided that either of the following sets of conditions is satisfied:

• (a)

On the interval $0, $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $x\ifrac{\mathrm{d}(x^{-1}g(x))}{\mathrm{d}x}$ is absolutely integrable.

• (b)

$g(x)$ is piecewise continuous and of bounded variation on every compact interval in $(0,\infty)$, and each of the following integrals

 10.43.32 $\int_{0}^{\frac{1}{2}}\frac{g(x)}{x}\ln\left(\frac{1}{x}\right)\mathrm{d}x,$ $\int_{\frac{1}{2}}^{\infty}\frac{|g(x)|}{x^{\frac{1}{2}}}\mathrm{d}x,$
• converges.

For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996).

For collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5).

## §10.43(vi) Compendia

For collections of integrals of the functions $I_{\nu}\left(z\right)$ and $K_{\nu}\left(z\right)$, including integrals with respect to the order, see Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).