# fundamental theorem of calculus

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##### 2: 28.29 Definitions and Basic Properties
###### §28.29(ii) Floquet’s Theorem and the Characteristic Exponent
28.29.4 $w_{\mbox{\tiny I}}(z+\pi,\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda),$
28.29.5 $w_{\mbox{\tiny II}}(z+\pi,\lambda)=w_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda).$
If $\nu$ $(\neq 0,1)$ is a solution of (28.29.9), then $F_{\nu}(z)$, $F_{-\nu}(z)$ comprise a fundamental pair of solutions of Hill’s equation. …
28.29.15 $\bigtriangleup(\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda)$
##### 3: 28.2 Definitions and Basic Properties
(28.2.1) possesses a fundamental pair of solutions $w_{\mbox{\tiny I}}(z;a,q),w_{\mbox{\tiny II}}(z;a,q)$ called basic solutions with …
28.2.6 $\mathscr{W}\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}\right\}=1,$
###### §28.2(iii) Floquet’s Theorem and the Characteristic Exponents
If $q\neq 0$, 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
###### §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 …
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)). …
##### 6: Bibliography K
• V. Kac and P. Cheung (2002) Quantum Calculus. Universitext, Springer-Verlag, New York.
• Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series ${}_{r+3}F_{r+2}$ . Integral Transforms Spec. Funct. 24 (11), pp. 916–921.
• B. J. King and A. L. Van Buren (1973) A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.
• D. E. Knuth (1968) The Art of Computer Programming. Vol. 1: Fundamental Algorithms. 1st edition, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont.
• T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.
• ##### 7: Bibliography M
• J. E. Marsden and A. J. Tromba (1996) Vector Calculus. 4th edition, W. H. Freeman & Company, New York.
• D. W. Matula and P. Kornerup (1980) 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.
• K. S. Miller and B. Ross (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.
• S. C. Milne (1988) A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.
• L. M. Milne-Thomson (1933) The Calculus of Finite Differences. Macmillan and Co. Ltd., London.
• ##### 8: 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. …
##### 9: 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). …
##### 10: 24.17 Mathematical Applications
###### §24.17(iii) Number Theory
Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …