implicit function theorem

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1: 1.5 Calculus of Two or More Variables

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Implicit Function Theorem

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2: 18.36 Miscellaneous Polynomials

… ►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem,

A_{n}A_{n-1}C_{n}>0

for

n\geq 1

as per (18.2.9_5). … ►The possibility of generalization to

\alpha=-k

, for

k\in\mathbb{N}

, is implicit in the identity Szegő (1975, page 102), … ►Consider the weight function … ►and orthonormal with respect to the weight function … ►In §18.39(i) it is seen that the functions,

\sqrt{w(x)}\hat{H}_{n+3}\left(x\right)

, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …

3: 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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§1.4(vi) Taylor’s Theorem for Real Variables

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4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Functions

f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)

for which

\left\langle f-g,f-g\right\rangle=0

are identified with each other. … ►The implicit boundary conditions taken here are that the

\phi_{n}(x)

and

\phi_{n}^{\prime}(x)

vanish as

x\to\pm\infty

, which in this case is equivalent to requiring

\phi_{n}(x)\in L^{2}\left(X\right)

, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point. … ►Eigenfunctions corresponding to the continuous spectrum are non-

L^{2}

functions. … ►For this latter see Simon (1973), and Reinhardt (1982); wherein advantage is taken of the fact that although branch points are actual singularities of an analytic function, the location of the branch cuts are often at our disposal, as they are not singularities of the function, but simply arbitrary lines to keep a function single valued, and thus only singularities of a specific representation of that analytic function. … ►Surprisingly, if

q(x)<0

on any interval on the real line, even if positive elsewhere, as long as

\int_{X}q(x)\,\mathrm{d}x\leq 0

, see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding

L^{2}\left(X\right)

eigenfunction. …