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Complex series and connected sets

100%

Jasek B.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS PREFACE..........................................................................................................................................................................3 INTRODUCTION............................................................................................................................................................. 4 1. Notation. 2. Subject of the paper. Chapter I. DECOMPOSITION OF Σ INTO $Σ_1$, $Σ_2$, $Σ_3$, $Σ_4$ INESSENTIAL RESTRICTION OF GENERALITY ............................................................................................................................................................ 6 1. 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....................................................................................................... 10 1. Chains of order n. 2. Further notations. 3. A sufficient condition for Ʌ(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 $S∈∑^0_1$................................................................................................ 20 1. Case: $S∈∑^0_4$. 2. Case: $S∈∑^0_1$. Chapter IV. CASES: $S∈∑^0_2$ and $S∈∑^0_3$ FAMILIES ɸ(S)..................................................................... 22 1. Notations. 2. Preliminary remarks on ɸ(S) for S from $∑^0_2$. 3. General theorems on ɸ(S) for S from $∑^0_2◡∑^0_3$. 4. Detailed remarks on ɸ(S). 5. The structure of $ɸ_0(S)$ for a special S from $∑^0_3$ Chapter V. CASE: $S∈∑^0_3$, FAMILIES Ω(S)...................................................................................................... 34 1. 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 for L(S) = C in the case $S∈Ω_4$. 4. Regions F_j(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........................................................................... 42 1. 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

2

On a certain transformation of complex series

37%

Jasek B.

Annales Polonici Mathematici

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1969-1970

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tom 22

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nr 3

277-289

3

Über Umordnung von Reihen

25%

Jasek B.

Colloquium Mathematicum

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1959-1960

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tom 7

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nr 2

257-259

4

Transformations of complex series

25%

Jasek B.

Colloquium Mathematicum

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1962

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tom 9

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nr 2

265-275

Ograniczanie wyników

2 Colloquium Mathematicum

1 Annales Polonici Mathematici

4 Jasek B.

1 1970

1 1966

1 1962

1 1960

Complex series and connected sets

On a certain transformation of complex series

Über Umordnung von Reihen

Transformations of complex series