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mean value theorems

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1: 1.4 Calculus of One Variable
Mean Value Theorem
First Mean Value Theorem
Second Mean Value Theorem
2: 1.6 Vectors and Vector-Valued Functions
Note that C can be given an orientation by means of 𝐜 . …
Green’s Theorem
Stokes’s Theorem
Gauss’s (or Divergence) Theorem
Green’s Theorem (for Volume)
3: Bille C. Carlson
In theoretical physics he is known for the “Carlson-Keller Orthogonalization”, published in 1957, Orthogonalization Procedures and the Localization of Wannier Functions, and the “Carlson-Keller Theorem”, published in 1961, Eigenvalues of Density Matrices. … Also, the homogeneity of the R -function has led to a new type of mean value for several variables, accompanied by various inequalities. …
4: 1.9 Calculus of a Complex Variable
DeMoivre’s Theorem
Cauchy’s Theorem
Liouville’s Theorem
Mean Value Property
Dominated Convergence Theorem
5: 2.1 Definitions and Elementary Properties
(In other words = here really means .) … For example, if f ( z ) is analytic for all sufficiently large | z | in a sector 𝐒 and f ( z ) = O ( z ν ) as z in 𝐒 , ν being real, then f ( z ) = O ( z ν 1 ) as z in any closed sector properly interior to 𝐒 and with the same vertex (Ritt’s theorem). This result also holds with both O ’s replaced by o ’s. … means that for each n , the difference between f ( x ) and the n th partial sum on the right-hand side is O ( ( x c ) n ) as x c in 𝐗 . … As an example, in the sector | ph z | 1 2 π δ ( < 1 2 π ) each of the functions 0 , e z , and e z (principal value) has the null asymptotic expansion …
6: 19.36 Methods of Computation
Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … All cases of R F , R C , R J , and R D are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). … This method loses significant figures in ρ if α 2 and k 2 are nearly equal unless they are given exact values—as they can be for tables. … When the values of complete integrals are known, addition theorems with ψ = π / 2 19.11(ii)) ease the computation of functions such as F ( ϕ , k ) when 1 2 π ϕ is small and positive. …These special theorems are also useful for checking computer codes. …
7: 10.18 Modulus and Phase Functions
10.18.17 M ν 2 ( x ) 2 π x ( 1 + 1 2 μ 1 ( 2 x ) 2 + 1 3 2 4 ( μ 1 ) ( μ 9 ) ( 2 x ) 4 + 1 3 5 2 4 6 ( μ 1 ) ( μ 9 ) ( μ 25 ) ( 2 x ) 6 + ) ,
10.18.18 θ ν ( x ) x ( 1 2 ν + 1 4 ) π + μ 1 2 ( 4 x ) + ( μ 1 ) ( μ 25 ) 6 ( 4 x ) 3 + ( μ 1 ) ( μ 2 114 μ + 1073 ) 5 ( 4 x ) 5 + ( μ 1 ) ( 5 μ 3 1535 μ 2 + 54703 μ 3 75733 ) 14 ( 4 x ) 7 + .
In (10.18.17) and (10.18.18) the remainder after n terms does not exceed the ( n + 1 ) th term in absolute value and is of the same sign, provided that n > ν 1 2 for (10.18.17) and 3 2 ν 3 2 for (10.18.18).