30Bxx Series expansions
This subtopic studies series expansions of complex functions, including power series, Laurent series, and convergence behavior around singularities.
Specific topics
30B10 Power series (including lacunary series) in one complex variable
Overview
30B10 studies power series (including lacunary series) in one complex variable in series expansions. It studies expansions of functions into power, Laurent, Fourier, and related series, with emphasis on convergence and summation.
Related Wikipedia Page
Series Expansions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for power series (including lacunary series) in one complex variable
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to approximate functions and analyze analytic structure.
Applications
- Fourier and power series
- Summability and convergence
- Approximation by series
References
Recommended Textbooks
30B20 Random power series in one complex variable
Overview
30B20 studies random power series in one complex variable in series expansions. It studies expansions of functions into power, Laurent, Fourier, and related series, with emphasis on convergence and summation.
Related Wikipedia Page
Series Expansions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for random power series in one complex variable
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to approximate functions and analyze analytic structure.
Applications
- Fourier and power series
- Summability and convergence
- Approximation by series
References
Recommended Textbooks
30B30 Boundary behavior of power series in one complex variable; over-convergence
Overview
30B30 studies boundary behavior of power series in one complex variable; over-convergence in series expansions. It studies expansions of functions into power, Laurent, Fourier, and related series, with emphasis on convergence and summation.
Related Wikipedia Page
Series Expansions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for boundary behavior of power series in one complex variable; over-convergence
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to approximate functions and analyze analytic structure.
Applications
- Fourier and power series
- Summability and convergence
- Approximation by series
References
Recommended Textbooks
30B40 Analytic continuation of functions of one complex variable
Overview
30B40 studies analytic continuation of functions of one complex variable in series expansions. It studies expansions of functions into power, Laurent, Fourier, and related series, with emphasis on convergence and summation.
Related Wikipedia Page
Series Expansions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for analytic continuation of functions of one complex variable
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to approximate functions and analyze analytic structure.
Applications
- Fourier and power series
- Summability and convergence
- Approximation by series
References
Recommended Textbooks
30B50 Dirichlet series, exponential series and other series in one complex variable
Overview
30B50 studies dirichlet series, exponential series and other series in one complex variable in series expansions. It studies expansions of functions into power, Laurent, Fourier, and related series, with emphasis on convergence and summation.
Related Wikipedia Page
Series Expansions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for dirichlet series, exponential series and other series in one complex variable
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to approximate functions and analyze analytic structure.
Applications
- Fourier and power series
- Summability and convergence
- Approximation by series
References
Recommended Textbooks
30B60 Completeness problems, closure of a system of functions
Overview
30B60 studies completeness problems, closure of a system of functions in series expansions. It studies expansions of functions into power, Laurent, Fourier, and related series, with emphasis on convergence and summation.
Related Wikipedia Page
Series Expansions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for completeness problems, closure of a system of functions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to approximate functions and analyze analytic structure.
Applications
- Fourier and power series
- Summability and convergence
- Approximation by series
References
Recommended Textbooks
30B70 Continued fractions; complex-analytic aspects
Overview
30B70 studies continued fractions; complex-analytic aspects in series expansions. It studies expansions of functions into power, Laurent, Fourier, and related series, with emphasis on convergence and summation.
Related Wikipedia Page
Series Expansions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for continued fractions; complex-analytic aspects
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to approximate functions and analyze analytic structure.
Applications
- Fourier and power series
- Summability and convergence
- Approximation by series
References
Recommended Textbooks