fundamental theorem of calculus

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1: 1.4 Calculus of One Variable

§1.4 Calculus of One Variable

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Mean Value Theorem

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Fundamental Theorem of Calculus

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First Mean Value Theorem

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Second Mean Value Theorem

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2: 28.29 Definitions and Basic Properties

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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

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28.29.4 $$w_{\text{I}}(z+\pi ,\lambda )=w_{\text{I}}(\pi ,\lambda )w_{\text{I}}(z,\lambda )+w_{\text{I}}^{\prime }(\pi ,\lambda )w_{\text{II}}(z,\lambda ),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E4

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See also:

Annotations for §28.29(ii), §28.29 and Ch.28

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28.29.5 $$w_{\text{II}}(z+\pi ,\lambda )=w_{\text{II}}(\pi ,\lambda )w_{\text{I}}(z,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda )w_{\text{II}}(z,\lambda ).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E5

Encodings:

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See also:

Annotations for §28.29(ii), §28.29 and Ch.28

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28.29.8

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E8

Encodings:

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See also:

Annotations for §28.29(ii), §28.29 and Ch.28

… ►If $\nu$ $(\ne 0,1)$ is a solution of (28.29.9), then $F_{\nu }(z)$, $F_{-\nu }(z)$ comprise a fundamental pair of solutions of Hill’s equation. …

3: 28.2 Definitions and Basic Properties

… ►(28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with … ►

28.2.6 $$𝒲\left\{w_{\text{I}},w_{\text{II}}\right\}=1,$$

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Symbols:

$𝒲$: Wronskian, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.2.E6

Encodings:

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See also:

Annotations for §28.2(ii), §28.2 and Ch.28

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§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

… ►If $q\ne 0$, then for a given value of $\nu$ the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …

4: 27.2 Functions

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§27.2(i) Definitions

►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►If $\left(a,n\right)=1$, then the Euler–Fermat theorem states that …

5: Bibliography M

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Vector Calculus. 4th edition, W. H. Freeman & Company, New York.

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Notes:

Fifth edition printed in 2003.

External Links:

ISBN 0-7167-2445-6, ZentralBlatt

Referenced by:

§1.5(i), §1.5(iii), §1.5(v), §1.5(vi), §1.6(i), §1.6(iii), §1.6(iv), §1.6(v).

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BibTeX

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Foundations of Finite Precision Rational Arithmetic. In Fundamentals of Numerical Computation (Computer-oriented Numerical Analysis), G. Alefeld and R. D. Grigorieff (Eds.), Comput. Suppl., Vol. 2, Vienna, pp. 85–111.

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External Links:

MathReview (G. Blanch), ZentralBlatt

Referenced by:

§3.1(iii).

Encodings:

BibTeX

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

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The Calculus of Finite Differences. Macmillan and Co. Ltd., London.

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Notes:

Reprinted in 1951 by Macmillan, and in 1981 by Chelsea Publishing Co., New York, and American Mathematical Society, AMS Chelsea Book Series, Providence, RI.

External Links:

ZentralBlatt

Referenced by:

§24.16(i), §24.17(i), §24.2(i), §24.2(ii), §24.4(i), §24.4(ii), §24.4(iii), §24.4(v), §26.1, §26.1, Notations B, Notations B.

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CODATA recommended values of the fundamental physical constants: 2002. Rev. Mod.Phys. 77, pp. 1–107.

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Referenced by:

§18.39(ii).

Encodings:

BibTeX

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6: Bibliography G

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Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.

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External Links:

MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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Applications of the classical umbral calculus. Algebra Universalis 49 (4), pp. 397–434.

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External Links:

ISSN 0002-5240, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§24.4(viii).

Encodings:

BibTeX

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A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.

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External Links:

ISSN 0002-9890, FullText, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.10(iv), §24.16(iii).

Encodings:

BibTeX

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Quantum mechanics: fundamentals. Second edition, Springer-Verlag, New York.

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External Links:

ISBN 0-387-22023-2, MathReview (Howard E. Brandt), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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7: Bibliography K

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Quantum Calculus. Universitext, Springer-Verlag, New York.

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External Links:

ISBN 0-387-95341-8, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

Ch.17.

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BibTeX

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An extension of Saalschütz’s summation theorem for the series ${}_{r+3}{}^{}F_{r+2}^{}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt

Referenced by:

§16.4(ii), §16.4(ii).

Encodings:

BibTeX

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A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.10.

Encodings:

BibTeX

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The Art of Computer Programming. Vol. 1: Fundamental Algorithms. 1st edition, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont.

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Notes:

Second printing of second edition by Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam, 1975.

External Links:

MathReview (M. Muller), ZentralBlatt

Referenced by:

§1.4(iii), (25.11.33).

Encodings:

BibTeX

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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8: 28.27 Addition Theorems

§28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

9: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

10: Tom M. Apostol

… ►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …