# kernel equations

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##### 1: 31.10 Integral Equations and Representations
and the kernel $\mathcal{K}(z,t)$ is a solution of the partial differential equation
###### Kernel Functions
31.10.18 $\frac{{\partial}^{2}\mathcal{K}}{{\partial u}^{2}}+\frac{{\partial}^{2}% \mathcal{K}}{{\partial v}^{2}}+\frac{{\partial}^{2}\mathcal{K}}{{\partial w}^{% 2}}+\frac{2\gamma-1}{u}\frac{\partial\mathcal{K}}{\partial u}+\frac{2\delta-1}% {v}\frac{\partial\mathcal{K}}{\partial v}+\frac{2\epsilon-1}{w}\frac{\partial% \mathcal{K}}{\partial w}=0.$
##### 4: 28.32 Mathematical Applications
Kernels $K$ can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. …
##### 5: 1.18 Linear 2nd Order Differential Operators and Eigenfunction Expansions
in which the integral kernel is defined asthis being the Dirac delta-function. Equation (1.18.29) often being called the completeness relation. The analogous orthonormality beingEquations (1.18.29) and (1.18.30) are usefully compared to equations (1.18.36) and (1.18.35), respectively. The operator itself, again as an integral kernel, may then be represented in terms of its eigenvalues and eigenfunctions:The general eigenfunction expansion of equation (1.18.37), then becomes the Fourier-Bessel eigenfunction expansion:
##### 6: 10.63 Recurrence Relations and Derivatives
Equations (10.63.6) and (10.63.7) also hold when the symbols $\operatorname{ber}$ and $\operatorname{bei}$ in (10.63.5) are replaced throughout by $\operatorname{ker}$ and $\operatorname{kei}$, respectively. …
##### 7: 10.68 Modulus and Phase Functions
Equations (10.68.8)–(10.68.14) also hold with the symbols $\operatorname{ber}$, $\operatorname{bei}$, $M$, and $\theta$ replaced throughout by $\operatorname{ker}$, $\operatorname{kei}$, $N$, and $\phi$, respectively. …
##### 8: 13.27 Mathematical Applications
The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. … For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).
##### 9: 10.61 Definitions and Basic Properties
When $\nu=0$ suffices on $\operatorname{ber}$, $\operatorname{bei}$, $\operatorname{ker}$, and $\operatorname{kei}$ are usually suppressed. Most properties of $\operatorname{ber}_{\nu}x$, $\operatorname{bei}_{\nu}x$, $\operatorname{ker}_{\nu}x$, and $\operatorname{kei}_{\nu}x$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter.
###### §10.61(ii) Differential Equations
$\operatorname{ker}_{-\nu}x=\cos\left(\nu\pi\right)\operatorname{ker}_{\nu}x-% \sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x,$
$\operatorname{ker}_{-n}x=(-1)^{n}\operatorname{ker}_{n}x,~{}\operatorname{kei}% _{-n}x=(-1)^{n}\operatorname{kei}_{n}x.$
##### 10: Bibliography T
• Y. Takei (1995) On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.
• S. A. Teukolsky (1972) Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.
• O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
• C. A. Tracy and H. Widom (1994) Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.
• J. F. Traub (1964) Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..