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1: 10.17 Asymptotic Expansions for Large Argument

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10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

$R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.17(iv), Erratum (V1.0.10) for Equation (10.17.14)

Permalink:

http://dlmf.nist.gov/10.17.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

►where $𝒱$ denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that $|\mathrm{\Im }t|$ changes monotonically. Bounds for ${𝒱}_{z,\mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by ►

10.17.15

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable and $\chi (\mathrm{\ell })$

Referenced by:

§10.17(iv)

Permalink:

http://dlmf.nist.gov/10.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

… ►The bounds (10.17.15) also apply to ${𝒱}_{z,-\mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ in the conjugate sectors. …

2: 10.40 Asymptotic Expansions for Large Argument

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10.40.11 $$|R_{\mathrm{\ell }}(\nu ,z)|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

$R_{\mathrm{\ell }}(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.40.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

►where $𝒱$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\mathrm{\Re }t|$ changes monotonically. Bounds for ${𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by ►

10.40.12

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, ${𝒱}_{a,b}\left(f\right)$: total variation and $z$: complex variable

Referenced by:

§10.40(i), §10.40(iii), Erratum (V1.0.11) for Equation (10.40.12)

Permalink:

http://dlmf.nist.gov/10.40.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.0.11):

Originally the third constraint was written incorrectly as $\pi \le |\mathrm{ph}z|\le \frac{3}{2}\pi$.

Suggested 2014-11-05 by Gergő Nemes

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

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3: 1.4 Calculus of One Variable

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1.4.33 $${𝒱}_{a,b}\left(f\right)=sup\sum _{j=1}^n\left|f(x_j)-f(x_{j-1})\right|,$$

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Defines:

$𝒱\left(f\right)$: total variation and ${𝒱}_{a,b}\left(f\right)$: total variation

Symbols:

$sup$: least upper bound (supremum), $j$: integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►If , then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)={𝒱}_{a,x}\left(f\right)$ and $h(x)={𝒱}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$. … ►

1.4.34 $${𝒱}_{a,b}\left(f\right)={\int }_a^b\left|f^{\prime }(x)\right|dx,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.4(v), §1.4(v), Erratum (V1.0.28) for Equation (1.4.34)

Permalink:

http://dlmf.nist.gov/1.4.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►Lastly, whether or not the real numbers $a$ and $b$ satisfy , and whether or not they are finite, we define ${𝒱}_{a,b}\left(f\right)$ by (1.4.34) whenever this integral exists. …

4: 2.7 Differential Equations

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2.7.23 $$|{ϵ}_j(x)|,\frac{1}{2}{f}^{-1/2}(x)|{ϵ}_j^{\prime }(x)|\le \exp \left(\frac{1}{2}{𝒱}_{a_j,x}\left(F\right)\right)-1,$$ $j=1,2$,

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Symbols:

$\exp z$: exponential function, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval, ${ϵ}_j(x)$: function and $F$: error-control function

Permalink:

http://dlmf.nist.gov/2.7.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

►provided that . …and $𝒱$ denotes the variational operator (§2.3(i)). … ►

2.7.25 $${𝒱}_{a_j,x}\left(F\right)=\left|{\int }_{a_j}^x\left|\frac{1}{f^{1/4}(t)}\frac{d^2}{{dt}^2}\left(\frac{1}{f^{1/4}(t)}\right)-\frac{g(t)}{f^{1/2}(t)}\right|dt\right|.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval and $F$: error-control function

Referenced by:

Erratum (V1.1.2) for Equation (2.7.25)

Permalink:

http://dlmf.nist.gov/2.7.E25

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value.

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

►Assuming also , we have …

5: 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. …

6: 2.3 Integrals of a Real Variable

… ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by ►

2.3.6 $${𝒱}_{a,b}\left(f(t)\right)={\int }_a^b\left|f^{\prime }(t)\right|dt;$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $a$: left endpoint and $b$: right endpoint

Referenced by:

§10.17(iv), Erratum (V1.1.1) for Equation (2.3.6)

Permalink:

http://dlmf.nist.gov/2.3.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.1):

The integrand was corrected so that the absolute value does not include the differential.

Suggested 2021-02-08 by Juan Luis Varona

See also:

Annotations for §2.3(i), §2.3 and Ch.2

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7: Errata

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2.7.25 $${𝒱}_{a_j,x}\left(F\right)=\left|{\int }_{a_j}^x\left|\frac{1}{f^{1/4}(t)}\frac{d^2}{{dt}^2}\left(\frac{1}{f^{1/4}(t)}\right)-\frac{g(t)}{f^{1/2}(t)}\right|dt\right|$$

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value for ${𝒱}_{a_j,x}\left(F\right)$.

… ►

2.3.6 $${𝒱}_{a,b}\left(f(t)\right)={\int }_a^b\left|f^{\prime }(t)\right|dt$$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Juan Luis Varona on 2021-02-08

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1.4.34 $${𝒱}_{a,b}\left(f\right)={\int }_a^b\left|f^{\prime }(x)\right|dx$$

The integrand has been corrected so that the absolute value does not include the differential.

Reported by Tran Quoc Viet on 2020-08-11

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10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right)$$

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes.

…

8: Bibliography S

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Variational calculations of scattering. Ann. Phys. 16, pp. 36–50.

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Referenced by:

§18.39(iv).

Encodings:

BibTeX

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An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.

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Notes:

This is a reprint, with corrections, of the first edition [University of Pennsylvania Press, Philadelphia, Pa., 1924].

External Links:

MathReview, ZentralBlatt

Referenced by:

§24.1, §24.6.

Encodings:

BibTeX

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The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.

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External Links:

Document, MathReview Entry

Referenced by:

§1.18(viii).

Encodings:

BibTeX

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Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

Encodings:

BibTeX

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9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Bounded and Unbounded Linear Operators

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Self-Adjoint Operators

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Spectrum of an Operator

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10: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►Analogous to (18.39.7) the 3D Schrödinger operator is …where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►The radial operator (18.39.28) … ►While $s$ in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b). …