analytically soluble examples

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11: 1 Algebraic and Analytic Methods

Chapter 1 Algebraic and Analytic Methods

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12: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. … ► $\ln z$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },0]$ and real-valued when $z$ ranges over the positive real numbers. … ►For another example see (4.2.37). … ►

§4.2(iii) The Exponential Function

… ►Another example of a principal value is provided by …

13: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. … ►For example, if in ${\text{P}}_{\text{II}}$ …

14: 2.7 Differential Equations

… ►is one at which the coefficients $f(z)$ and $g(z)$ are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution $w_j(z)$ can be continued analytically into any other sector. … ►In a finite or infinite interval $(a_1,a_2)$ let $f(x)$ be real, positive, and twice-continuously differentiable, and $g(x)$ be continuous. … ►

Example

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15: 36.12 Uniform Approximation of Integrals

… ►Also, $f$ is real analytic, and ${\partial }^{K+2}f/{\partial u}^{K+2}>0$ for all $𝐲$ such that all $K+1$ critical points coincide. … ►The square roots are real and positive when $𝐲$ is such that all the critical points are real, and are defined by analytic continuation elsewhere. The quantities $a_m(𝐲)$ are real for real $𝐲$ when $g$ is real analytic. … ►For details of this example, see Paris (1991). … ►The coefficients of $\mathrm{Ai}$ and ${\mathrm{Ai}}^{\prime }$ are real if $y$ is real and $g$ is real analytic. …

16: 2.10 Sums and Sequences

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Example

… ►In the present example it leads to … ►

Example

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Example

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Example

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17: 33.23 Methods of Computation

… ►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … ►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …

18: Frank W. J. Olver

… ►Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►

1 Algebraic and Analytic Methods

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19: 29 Lamé Functions

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20: Simon Ruijsenaars

… ►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …