Entire Functions of Several Complex Variables - In-depth

In this review of Entire Functions of Several Complex Variables the reviewer finds a focused, scholarly treatment intended for mathematicians and advanced students who need a rigorous reference on growth conditions and applications. The book is best for readers working in complex analysis, PDEs via Fourier-Laplace transforms, analytic number theory, or approximation theory who require precise statements about entire functions in several variables and constructions of solutions with controlled growth.

Key Features

• Comprehensive focus: The text concentrates on entire functions of several complex variables and explains why they form an original chapter in complex analysis, giving readers a clear thematic scope.
• Applications oriented: The material links theory to applications such as partial differential equations, analytic number theory, and approximation theory, which helps readers apply results beyond pure abstraction.
• Growth condition emphasis: The book emphasizes solving problems under specified growth scales, enabling readers to locate solutions with prescribed asymptotic behavior or optimal growth.
• Problem-driven approach: Examples and motivations come from concrete problems like convolution operators and the Fourier-Laplace transform, clarifying why particular results are useful.
• Rigorous presentation: The authorial approach is aimed at researchers needing precise, mathematically rigorous statements rather than informal exposition, supporting advanced study and citation.

Who It's For

Entire Functions of Several Complex Variables is best suited to graduate students, researchers, and practitioners in pure mathematics who already have a background in complex analysis and want a concentrated reference on several complex variables and growth-related existence results. Those working on PDEs, analytic number theory, or approximation problems who need solution existence with growth constraints will find it particularly valuable.

Readers seeking a gentle introduction, extensive exercises for beginners, or a broad textbook-style survey should look elsewhere, as this work prioritizes depth and applications over pedagogical hand-holding or elementary motivation.

Pros & Cons

Pros

• Authoritative treatment focused on an important, original subfield of complex analysis.
• Clear links to practical applications such as PDEs via the Fourier-Laplace transform and convolution operators.
• Useful for finding solutions that satisfy explicit growth conditions and for studying optimal asymptotic behavior.

Cons

• Not designed as an introductory textbook; it assumes prior familiarity with advanced complex analysis.

Specifications

Title	Entire Functions of Several Complex Variables
Series	Grundlehren der mathematischen Wissenschaften
Author / Brand	Pierre LelongLawrence Gruman
Primary Topic	Entire functions in several complex variables
Applications Highlighted	Partial differential equations, analytic number theory, approximation theory
Key Method	Fourier-Laplace transformation and convolution operators

Our Verdict

For advanced students and researchers who need a concentrated, rigorous reference on entire functions in several complex variables and on constructing solutions with controlled growth, this work delivers strong value. It connects theoretical results to applications in PDEs and number theory, making it a useful addition to a specialist mathematical library.

Frequently Asked Questions

Does this book cover applications to partial differential equations?
Yes. The review notes explicit connections to PDEs via the Fourier-Laplace transform and convolution operators.

Is this suitable as a first course text?
No. The material is advanced and better suited to readers with prior complex analysis experience rather than beginners.

Does it address growth conditions for solutions?
Yes. A central theme is finding solutions that satisfy specific growth scales and sometimes minimal asymptotic growth.