March 7, 2017

Analytic functions and distributions in physics and by Bernard W. Roos

By Bernard W. Roos

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19) transformed into the new coordinates for -T ~ 8 ~ T, y E il', where il' is a neighborhood of zero in ~-1 and T > 0 is sufficiently small; moreover v(s, y) == 0 for o ~ 8 ~ Iyl (cf. 3b). 19) has the same form, but with different matrices Aj and B. Therefore we may assume from the outset that u(t,x) == 0 for t ~ Ixl 2 and it is not necessary to introduce the new variables 8 and y. Consider the "conjugate" system of equations 0:: = t j=l o~. (Aj(t,x)1/J) 3 . 21) 36 Chapter 1. Basic Concepts t b a Fig.

Sts C such that ImAj(e) > -C for e E JRn, j = 1, ... , m. 43) We denote by Pm the leading homogeneous part of the polynomial P, also called the principal symbol 01 the operator p(Dt , Dz). To be specific if L p(7J) = Pa7Ja lal~m (here a is an (n + 1)-dimensional multi-index and 7J = (T, Pm(7J) = L e», then Pa7J a . 43) for P a similar condition follows for Pm, which, as is easily seen, is equivalent to the condition that all the roots of the equation Pm (T, = 0 with respect to T for E an are real.

In this situation the 6 found and, consequently, the radius of convergence of the power series giving the solution u will be independent of the right-hand side f (provided it is an entire function). tu\ Zn=O = If'i(x), j = O,I, ... 8. 6); here it is necessary to assume that the data If'i are holomorphic and bounded in the polydisk [JR={zECn-1:lzil and then 6 = 6(R) independent of f and If'j (but depending on the operator A) such that 31 §2.

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