Complex series and connected sets

B. Jasek

GDML_Books, (1966), p.

Résumé

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO $Σ_1$, $Σ_2$, $Σ_3$, $Σ_4$ INESSENTIAL RESTRICTIONOF GENERALITY ............................................................................................................................................................ 61. Families $Σ_k$, k = 1, 2, 3, 4. 2. Families $Σ^0$ and $Σ^0_k$, k = 1, 2, 3, 4.Chapter II. FURTHER AUXILIARY THEOREMS....................................................................................................... 101. Chains of order n. 2. Further notations. 3. A sufficient condition forɅ(S) = Γ. Property (.). 4. A lemma on complex numbers. 5. Properties(..), (...) and (....). 6 A necessary and sufficient condition for Ʌ (S) = Γ.Chapter III. CASES: $S∈∑^0_4$ and $S∈∑^0_1$................................................................................................ 201. Case: $S∈∑^0_4$. 2. Case: $S∈∑^0_1$.Chapter IV. CASES: $S∈∑^0_2$ and $S∈∑^0_3$ FAMILIES ɸ(S)..................................................................... 221. Notations. 2. Preliminary remarks on ɸ(S) for S from $∑^0_2$. 3. Generaltheorems on ɸ(S) for S from $∑^0_2◡∑^0_3$. 4. Detailed remarks on ɸ(S). 5. Thestructure of $ɸ_0(S)$ for a special S from $∑^0_3$Chapter V. CASE: $S∈∑^0_3$, FAMILIES Ω(S)...................................................................................................... 341. Definitions of the families Ω, Ω(S), $Ω_k$ and $Ω_k(S)$, k = 0, 1, 2, 3, 4.2. Families $Ω^n_k$, k = 0, 1, 2, 3, 4 and $Ω^n$. 3. A sufficient condition forL(S) = C in the case $S∈Ω_4$. 4. Regions Fj(z, p; e), j = 1, 2, 3, 4. 5.Families $Ω_4(S)$. 6. Families $Ω_3(S)$ and Ω(S).Chapter VI. CASE: $S∈∑^0_2◡∑^0_3$ VARIOUS PROBLEMS........................................................................... 421. Property (—). 2. An example of the equality Λ(S) = Γ for S from $∑^0_3$3. An open problem concerning $Λ_0(S)$REFERENCES................................................................................................................................................................ 46

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Zbl 0166.31702

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