reparametrization of integration paths

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21: 6.12 Asymptotic Expansions

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22: 3.7 Ordinary Differential Equations

… ►Assume that we wish to integrate (3.7.1) along a finite path $𝒫$ from $z=a$ to $z=b$ in a domain $D$. The path is partitioned at $P+1$ points labeled successively $z_0,z_1,\mathrm{\dots },z_P$, with $z_0=a$, $z_P=b$. … ►Now suppose the path $𝒫$ is such that the rate of growth of $w(z)$ along $𝒫$ is intermediate to that of two other solutions. … ►The larger the absolute values of the eigenvalues ${\lambda }_k$ that are being sought, the smaller the integration steps $\left|{\tau }_j\right|$ need to be. …

23: 5.13 Integrals

… ►In (5.13.1) the integration path is a straight line parallel to the imaginary axis. …

24: 15.6 Integral Representations

… ►In all cases the integrands are continuous functions of $t$ on the integration paths, except possibly at the endpoints. … ►In (15.6.3) the point $1/(z-1)$ lies outside the integration contour, the contour cuts the real axis between $t=-1$ and $0$, at which point $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1+t\right)=0$. … ►In (15.6.5) the integration contour starts and terminates at a point $A$ on the real axis between $0$ and $1$. …If desired, and as in Figure 5.12.3, the upper integration limit in (15.6.5) can be replaced by $(1+,0+,1-,0-)$. However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by $-1$. …

25: 8.2 Definitions and Basic Properties

… ►without restrictions on the integration paths. However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ take their principal values; compare §4.2(i). …

26: Alexander A. Its

… ►Current research areas of Its are mathematical physics, special functions, and integrable systems. … ► Novokshënov), published by Springer in 1986, Algebro-geometric Approach to Nonlinear Integrable Problems (with E. …

27: Vadim B. Kuznetsov

… ►Kuznetsov published papers on special functions and orthogonal polynomials, the quantum scattering method, integrable discrete many-body systems, separation of variables, Bäcklund transformation techniques, and integrability in classical and quantum mechanics. …

28: 21.7 Riemann Surfaces

… ►Note that for the purposes of integrating these holomorphic differentials, all cycles on the surface are a linear combination of the cycles $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$. … ►where $P_1$ and $P_2$ are points on $\mathrm{\Gamma }$, $𝝎=({\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_g)$, and the path of integration on $\mathrm{\Gamma }$ from $P_1$ to $P_2$ is identical for all components. … ►where again all integration paths are identical for all components. …

29: 9.14 Incomplete Airy Functions

… ►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …

30: Alexander I. Bobenko

… ►Bobenko’s books are Algebro-geometric Approach to Nonlinear Integrable Problems (with E. … Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …He is also coeditor of Discrete Integrable Geometry and Physics (with R. …