Author | Bergman, Stefan. author |
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Title | Integral Operators in the Theory of Linear Partial Differential Equations [electronic resource] / by Stefan Bergman |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1961 |
Connect to | http://dx.doi.org/10.1007/978-3-642-64985-1 |
Descript | X, 148 p. online resource |
I. Differential equations in two variables with entire coefficients -- ยง 1. A representation of solutions of partial differential equations -- ยง 2. The integral operator of the first kind -- ยง 3. Further representations of integral operators -- ยง 4. A representation of the operator of the first kind in terms of integrals -- ยง 5. Properties of the integral operator of the first kind -- ยง 6. Some further properties of the integral operator of the first kind -- ยง 7. The differential equation ?2V + F(r2) V = 0 -- ยง 8. Integral operators of exponential type -- ยง 9. The differential equation ?2? + N (x) ? = 0 -- ยง 10. Differential equations of higher order -- II. Harmonic functions in three variables -- ยง 1. Preliminaries -- ยง 2. Characteristic space ?3 -- ยง 3. Harmonic functions with rational B3-associates -- ยง 4. Period functions -- ยง 5. Relations between coefficients of a series development of a harmonic function and its singularities -- ยง 6. Another type of integral representations of harmonic functions -- ยง 7. The behavior in the large of functions of the class S (E, ?0, ?1) with a rational associate f (?) -- III. Differential equations in three variables -- ยง 1. An integral operator generating solutions of the equation ?3? + A (r2) X ยท ? ? + C (r2) ? = 0 -- ยง 2. A series expansion for solutions of the equation ?3? + A (r2) X ยท ? ? + C (r2) ? = 0 -- ยง 3. An integral operator generating solutions of the equation ?3? + F (y, z) ? = 0 -- ยง 4. A second integral operator generating solutions of the equation ?3? + F (y, z) ? = 0 -- ยง 5. An integral operator generating solutions of the equation ?x + ?yy + ?zz + F (y, z) ? = 0 -- ยง 6. An integral operator generating solutions of the equation g???????+h????+k? = 0 -- IV. Systems of differential equations -- ยง 1. Harmonic vectors of three variables. Preliminaries -- ยง 2. Harmonic vectors in the large and their representation as integrals -- ยง 3. Integrals of harmonic vectors -- ยง 4. Relations between integrals of algebraic harmonic vectors in three variables and integrals of algebraic functions of a complex variable -- ยง 5. Generalization of the residue theorems to the case of the equation ?3? + F (r2) ? = 0 -- ยง 6. An operator generating solutions of a system of partial differential equations -- V. Equations of mixed type and elliptic equations with singular and non-analytic -- ยง 1. Introduction. The simplified case of equations of mixed type -- ยง 2. A generalization of the representation (1.12) of solutions of the equation (1.6) -- ยง 3. The operator (1.11b) in the general case -- ยง 4. Generating functions analogous to solutions of the hypergeometric equation -- ยง 5. On the solution of the initial value problem in the large -- ยง 6. Generalized Cauchy-Riemann equations -- ยง 7. The differential equation ?2? + N(x)? = 0 with a new type of singularity of N -- ยง 8. An integral operator for equations with non-analytic coefficients