expansions in partial fractions

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11: 2.4 Contour Integrals

… ►In consequence, the asymptotic expansion obtained from (2.4.14) is no longer null. … ►The final expansion then has the form … ►Suppose that on the integration path $𝒫$ there are two simple zeros of $\partial p(\alpha ,t)/\partial t$ that coincide for a certain value $\widehat{\alpha }$ of $\alpha$. … ►with $a$ and $b$ chosen so that the zeros of $\partial p(\alpha ,t)/\partial t$ correspond to the zeros $w_1(\alpha ),w_2(\alpha )$, say, of the quadratic $w^2+2aw+b$. … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …

12: 3.11 Approximation Techniques

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§3.11(ii) Chebyshev-Series Expansions

… ►the expansion …In fact, (3.11.11) is the Fourier-series expansion of $f(\cos \theta )$; compare (3.11.6) and §1.8(i). … ►However, in general (3.11.11) affords no advantage in $ℂ$ for numerical purposes compared with the Maclaurin expansion of $f(z)$. ►For further details on Chebyshev-series expansions in the complex plane, see Mason and Handscomb (2003, §5.10). …

13: 19.20 Special Cases

… ►In this subsection, and also §§19.20(ii)–19.20(v), the variables of all $R$-functions satisfy the constraints specified in §19.16(i) unless other conditions are stated. … ►When the variables are real and distinct, the various cases of $R_J(x,y,z,p)$ are called circular (hyperbolic) cases if $(p-x)(p-y)(p-z)$ is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. …Since , $p$ is in a hyperbolic region. In the complete case ($x=0$) (19.20.14) reduces to …

14: Errata

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A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

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In §15.6, it was noted that (15.6.8) can be rewritten as a fractional integral.

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A new Subsection Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

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A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

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36.10.14 $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0$$

Originally this equation appeared with $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

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15: 8.19 Generalized Exponential Integral

… ►For $p,z\in ℂ$ … ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

§8.19(iv) Series Expansions

… ►When $p\in ℂ$ … ►

§8.19(vii) Continued Fraction

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