analytically soluble examples

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1: 34.9 Graphical Method

… ►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. Thus, any analytic expression in the theory, for example equations (34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be represented by a diagram; conversely, any diagram represents an analytic equation. …For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

2: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …

3: 1.10 Functions of a Complex Variable

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Examples

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§1.10(ii) Analytic Continuation

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Analytic Functions

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Example

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4: 18.39 Applications in the Physical Sciences

… ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. Also presented are the analytic solutions for the $L^2$, bound state, eigenfunctions and eigenvalues of the Morse oscillator which also has analytically known non-normalizable continuum eigenfunctions, thus providing an example of a mixed spectrum. … ►

1D Quantum Systems with Analytically Known Stationary States

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Other Analytically Solved Schrödinger Equations

… ►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the $x_i$ and $w_i$, from the J-matrix as in §3.5(vi), straightforward. …

5: 1.13 Differential Equations

… ►The equation …where $z\in D$, a simply-connected domain, and $f(z)$, $g(z)$ are analytic in $D$, has an infinite number of analytic solutions in $D$. … ►Assume that in the equation … ►The inhomogeneous (or nonhomogeneous) equation …with $f(z)$, $g(z)$, and $r(z)$ analytic in $D$ has infinitely many analytic solutions in $D$. …

6: 2.4 Contour Integrals

… ►If $q(t)$ is analytic in a sector containing $\mathrm{ph}t=0$, then the region of validity may be increased by rotation of the integration paths. … ►is continuous in $\mathrm{\Re }z\ge c$ and analytic in $\mathrm{\Re }z>c$, and by inversion (§1.14(iii)) … ►Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion … ►Assume that $p(t)$ and $q(t)$ are analytic on an open domain $𝐓$ that contains $𝒫$, with the possible exceptions of $t=a$ and $t=b$. … ►For examples see Olver (1997b, Chapter 4). …

7: 3.8 Nonlinear Equations

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Example

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Example

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Example

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§3.8(v) Zeros of Analytic Functions

… ►Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …

8: 3.12 Mathematical Constants

… ►can be defined analytically in numerous ways, for example, …

9: 15.2 Definitions and Analytical Properties

§15.2 Definitions and Analytical Properties

… ►on the disk , and by analytic continuation elsewhere. … ►again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. … ►

§15.2(ii) Analytic Properties

… ►The right-hand side can be seen as an analytical continuation for the left-hand side when $a$ approaches $-m$. …

10: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. … ►

Example

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