Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |
From inside the book
Results 1-3 of 100
Page 122
... log(c2;t'+i), from which our claim follows. Thus we have that log(c3jto) - max(log(cijto), log(c2;to)) > 12fc, and log(r3)to) - max(log(rijt0), log(r2;t0)) < k. After another 2.1k timesteps, we have thatlog(?-3jt0) — max(log(rijto),log ...
... log(c2;t'+i), from which our claim follows. Thus we have that log(c3jto) - max(log(cijto), log(c2;to)) > 12fc, and log(r3)to) - max(log(rijt0), log(r2;t0)) < k. After another 2.1k timesteps, we have thatlog(?-3jt0) — max(log(rijto),log ...
Page 153
... R2 + ( е1 + € 2 ) / R2 ; ¥ 2 = ( e2 + € 1 ) / R3 ; and hence e1 = R2R1 / ( R2 – R1 ) . Y1 - R2 . R1 / ( R2 – R1 ) ... ( log R2 – log r ) / ( log R2 – log R1 ) - - + ¥ 2 . ( log r – log R1 ) / ( log R2 – log R1 ) . - For a point outside the ...
... R2 + ( е1 + € 2 ) / R2 ; ¥ 2 = ( e2 + € 1 ) / R3 ; and hence e1 = R2R1 / ( R2 – R1 ) . Y1 - R2 . R1 / ( R2 – R1 ) ... ( log R2 – log r ) / ( log R2 – log R1 ) - - + ¥ 2 . ( log r – log R1 ) / ( log R2 – log R1 ) . - For a point outside the ...
Page 246
... log r + log R2 - log R1 T1 log R2 T2 log R1 log R2 - log R1 2 ( 62.1 ) Hence , denoting by F ( z ) the same quantity as in § 46 and omitting the imaginary part of an arbitrary constant , one obtains 2 T2- T1 F ( z ) log z + log R2 - log R1 ...
... log r + log R2 - log R1 T1 log R2 T2 log R1 log R2 - log R1 2 ( 62.1 ) Hence , denoting by F ( z ) the same quantity as in § 46 and omitting the imaginary part of an arbitrary constant , one obtains 2 T2- T1 F ( z ) log z + log R2 - log R1 ...
Other editions - View all
Common terms and phrases
analytic annulus applying Green's B₂ Borel measures boundary component bounded convergence bounded sets characteristic convergence compact operator compact set compact subsets conclude connected admissible domains constant continuous extension continuous with respect convergence theorem D₁ D₂ defined definition denote dielectric dielectric Green's function disc of radius domain of given double-layer potentials eigen eigenfunction eigenfunctions associated eigenvalue and associated extremal problem feL D finite follows given transfinite diameter Green's function Green's identity harmonic Hence Hölder continuous implies kernel log|z monotonicity Moreover neighborhood non-negative normalized eigenfunction o(p² pointwise and boundedly proof satisfies Schwarz inequality simply connected slit plane solution spectrum subharmonic sufficiently small superharmonic Theorem 2.4 transfinite diameter uniform convergence uniformly unique negative eigenvalue vanish identically vanishes at infinity ατ πλ эко