# with respect to integration

(0.003 seconds)

## 1—10 of 36 matching pages

##### 3: 3.11 Approximation Techniques
They enjoy an orthogonal property with respect to integrals: …
##### 4: 36.2 Catastrophes and Canonical Integrals
with the contour passing to the lower right of $u=0$. …with the contour passing to the upper right of $u=0$. … $\Psi_{1}$ is related to the Airy function (§9.2): … … Addendum: For further special cases see §36.2(iv)
##### 5: 20.10 Integrals
###### §20.10(i) Mellin Transforms with respectto the Lattice Parameter
20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\mathrm{d}x=2^{s% }(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\mathrm{d}x=% \pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$
20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\mathrm{d}x=% (1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right).$
##### 6: 7.7 Integral Representations
In (7.7.13) and (7.7.14) the integration paths are straight lines, $\zeta=\frac{1}{16}\pi^{2}z^{4}$, and $c$ is a constant such that $0 in (7.7.13), and $0 in (7.7.14). …
##### 7: 2.8 Differential Equations with a Parameter
dots denoting differentiations with respect to $\xi$. … The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi$. … The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi$. … The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi$. … The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to $\xi$. …
##### 8: 2.3 Integrals of a Real Variable
###### §2.3(i) Integration by Parts
(In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) … derives from the neighborhood of the minimum of $p(t)$ in the integration range. … In consequence, the approximation is nonuniform with respect to $\alpha$ and deteriorates severely as $\alpha\to 0$. A uniform approximation can be constructed by quadratic change of integration variable: …
##### 9: 4.37 Inverse Hyperbolic Functions
Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. In (4.37.3) the integration path may not intersect $\pm 1$. … The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. … These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. … are respectively given by …
##### 10: 2.4 Contour Integrals
Then by integration by parts the integral … The most successful results are obtained on moving the integration contour as far to the left as possible. …
• (c)

Excluding $t=a$, $\Re\left(e^{i\theta}p(t)-e^{i\theta}p(a)\right)$ is positive when $t\in\mathscr{P}$, and is bounded away from zero uniformly with respect to $\theta\in[\theta_{1},\theta_{2}]$ as $t\to b$ along $\mathscr{P}$.

• The problem of obtaining an asymptotic approximation to $I(\alpha,z)$ that is uniform with respect to $\alpha$ in a region containing $\widehat{\alpha}$ is similar to the problem of a coalescing endpoint and saddle point outlined in §2.3(v). The change of integration variable is given by …