variation of parameters

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11: 2.8 Differential Equations with a Parameter

§2.8 Differential Equations with a Parameter

… ►In addition, ${𝒱}_{{𝒬}_j}\left(A_1\right)$ and ${𝒱}_{{𝒬}_j}\left(A_n\right)$ must be bounded on ${𝚫}_j({\alpha }_j)$. … ►These results are valid when ${𝒱}_{{\alpha }_1,{\alpha }_2}\left({|\xi |}^{1/2}{B}_0\right)$ and ${𝒱}_{{\alpha }_1,{\alpha }_2}\left({|\xi |}^{1/2}{B}_{n-1}\right)$ are finite. … ►These results are valid when ${𝒱}_{0,{\alpha }_2}\left({\xi }^{1/2}{B}_0\right)$ and ${𝒱}_{0,{\alpha }_2}\left({\xi }^{1/2}{B}_{n-1}\right)$ are finite. … ►These results are valid when ${𝒱}_{{\alpha }_1,0}\left({|\xi |}^{1/2}{B}_0\right)$ and ${𝒱}_{{\alpha }_1,0}\left({|\xi |}^{1/2}{B}_{n-1}\right)$ are finite. …

12: 10.43 Integrals

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10.43.7 $${\int }_0^x{\mathrm{e}}^{\pm t}{t}^{\nu }{I}_{\nu }\left(t\right)dt=\frac{{\mathrm{e}}^{\pm x}{x}^{\nu +1}}{2\nu +1}(I_{\nu }\left(x\right)\mp I_{\nu +1}\left(x\right)),$$ $\mathrm{\Re }\nu >-\frac{1}{2}$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.43.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(ii), §10.43 and Ch.10

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10.43.10 $${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^t{t}^{-\nu }{K}_{\nu }\left(t\right)dt=\frac{{\mathrm{e}}^x{x}^{-\nu +1}}{2\nu -1}(K_{\nu }\left(x\right)+K_{\nu -1}\left(x\right)),$$ $\mathrm{\Re }\nu >\frac{1}{2}$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.43(ii)

Permalink:

http://dlmf.nist.gov/10.43.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(ii), §10.43 and Ch.10

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10.43.18 $${\int }_0^{\mathrm{\infty }}{K}_{\nu }\left(t\right)dt=\frac{1}{2}\pi \sec \left(\frac{1}{2}\pi \nu \right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $\sec z$: secant function and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.43.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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10.43.23 $${\int }_0^{\mathrm{\infty }}{t}^{\nu +1}{I}_{\nu }\left(bt\right)\exp \left(-p^2{t}^2\right)dt=\frac{b^{\nu }}{{(2p^2)}^{\nu +1}}\exp \left(\frac{b^2}{4p^2}\right),$$ $\mathrm{\Re }\nu >-1,\mathrm{\Re }\left(p^2\right)>0$,

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.43.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §10.43(iv), §10.43 and Ch.10

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

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