# fundamental solutions

(0.001 seconds)

## 1—10 of 24 matching pages

##### 1: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …
##### 2: 28.29 Definitions and Basic Properties
28.29.4 $w_{\mbox{\tiny I}}(z+\pi,\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda),$
28.29.5 $w_{\mbox{\tiny II}}(z+\pi,\lambda)=w_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda).$
28.29.8 $\begin{bmatrix}w_{\mbox{\tiny I}}(\pi,\lambda)&w_{\mbox{\tiny II}}(\pi,\lambda% )\\ w^{\prime}_{\mbox{\tiny I}}(\pi,\lambda)&w^{\prime}_{\mbox{\tiny II}}(\pi,% \lambda)\end{bmatrix}.$
28.29.9 $2\cos\left(\pi\nu\right)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda).$
28.29.15 $\bigtriangleup(\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda)$
##### 3: 28.2 Definitions and Basic Properties
(28.2.1) possesses a fundamental pair of solutions $w_{\mbox{\tiny I}}(z;a,q),w_{\mbox{\tiny II}}(z;a,q)$ called basic solutions with
28.2.5 $\begin{bmatrix}w_{\mbox{\tiny I}}(0;a,q)&w_{\mbox{\tiny II}}(0;a,q)\\ w^{\prime}_{\mbox{\tiny I}}(0;a,q)&w^{\prime}_{\mbox{\tiny II}}(0;a,q)\end{% bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}.$
28.2.6 $\mathscr{W}\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}\right\}=1,$
28.2.7 $w_{\mbox{\tiny I}}(z\pm\pi;a,q)=w_{\mbox{\tiny I}}(\pi;a,q)w_{\mbox{\tiny I}}(% z;a,q)\pm w^{\prime}_{\mbox{\tiny I}}(\pi;a,q)w_{\mbox{\tiny II}}(z;a,q),$
28.2.8 $w_{\mbox{\tiny II}}(z\pm\pi;a,q)=\pm w_{\mbox{\tiny II}}(\pi;a,q)w_{\mbox{% \tiny I}}(z;a,q)+w^{\prime}_{\mbox{\tiny II}}(\pi;a,q)w_{\mbox{\tiny II}}(z;a,% q),$
##### 4: 15.10 Hypergeometric Differential Equation
###### §15.10(i) FundamentalSolutions
When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs $f_{1}(z)$, $f_{2}(z)$ of fundamental solutions. … (a) If $c$ equals $n=1,2,3,\dots$, and $a=1,2,\dots,n-1$, then fundamental solutions in the neighborhood of $z=0$ are given by (15.10.2) with the interpretation (15.2.5) for $f_{2}(z)$. … The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. …
##### 5: 16.8 Differential Equations
When no $b_{j}$ is an integer, and no two $b_{j}$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by … When $p=q+1$, and no two $a_{j}$ differ by an integer, another fundamental set of solutions of (16.8.3) is given by … In this reference it is also explained that in general when $q>1$ no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near $z=1$. …
##### 6: Bibliography V
• H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.
• ##### 7: 28.10 Integral Equations
28.10.9 $\int_{0}^{\ifrac{\pi}{2}}J_{0}\left(2\sqrt{q({\cos}^{2}\tau-{\sin}^{2}\zeta)}% \right)\mathrm{ce}_{2n}\left(\tau,q\right)\mathrm{d}\tau=w_{\mbox{\tiny II}}(% \tfrac{1}{2}\pi;a_{2n}\left(q\right),q)\mathrm{ce}_{2n}\left(\zeta,q\right),$
28.10.10 $\int_{0}^{\pi}J_{0}\left(2\sqrt{q}(\cos\tau+\cos\zeta)\right)\mathrm{ce}_{n}% \left(\tau,q\right)\mathrm{d}\tau=w_{\mbox{\tiny II}}(\pi;a_{n}\left(q\right),% q)\mathrm{ce}_{n}\left(\zeta,q\right).$
##### 8: 1.13 Differential Equations
###### Fundamental Pair
Two solutions $w_{1}(z)$ and $w_{2}(z)$ are called a fundamental pair if any other solution $w(z)$ is expressible as … If $w_{0}(z)$ is any one solution, and $w_{1}(z)$, $w_{2}(z)$ are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as …
##### 9: 28.4 Fourier Series
28.4.24 $\frac{A^{2n}_{2m}(q)}{A^{2n}_{0}(q)}=\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{q}{4% }\right)^{m}\frac{\pi\left(1+O\left(m^{-1}\right)\right)}{w_{\mbox{\tiny II}}(% \frac{1}{2}\pi;a_{2n}\left(q\right),q)},$
28.4.26 $\frac{B^{2n+1}_{2m+1}(q)}{B^{2n+1}_{1}(q)}=\frac{(-1)^{m}}{\left({\left(\tfrac% {1}{2}\right)_{m+1}}\right)^{2}}\left(\frac{q}{4}\right)^{m+1}\frac{2\left(1+O% \left(m^{-1}\right)\right)}{w_{\mbox{\tiny I}}(\frac{1}{2}\pi;b_{2n+1}\left(q% \right),q)},$
##### 10: 13.2 Definitions and Basic Properties
###### §13.2(v) Numerically Satisfactory Solutions
Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are … A fundamental pair of solutions that is numerically satisfactory near the origin is …