partial fractions

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21: 2.4 Contour Integrals

… ►However, if $p^{\prime }(t_0)=0$, then $\mu \ge 2$ and different branches of some of the fractional powers of $p_0$ are used for the coefficients $b_s$; again see §2.3(iii). … ►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 $w^2+2aw+b$. … ►

2.4.20 $$f(\alpha ,w)=q(\alpha ,t)\frac{dt}{dw}=q(\alpha ,t)\frac{w^2+2aw+b}{\partial p(\alpha ,t)/\partial t}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $p(t)$: analytic function, $q(t)$: analytic function, $w$: change of variable, $a$ , $b$ and $f(\alpha ,w)$: function

Permalink:

http://dlmf.nist.gov/2.4.E20

Encodings:

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See also:

Annotations for §2.4(v), §2.4 and Ch.2

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