entire function: meaning, definition, pronunciation and examples
Very Low Frequency / SpecializedFormal, Technical (exclusively academic mathematics)
Quick answer
What does “entire function” mean?
A complex-valued function that is holomorphic (complex-differentiable) at every point in the complex plane.
Audio
Pronunciation
Definition
Meaning and Definition
A complex-valued function that is holomorphic (complex-differentiable) at every point in the complex plane.
In mathematics, a function that is analytic everywhere except at infinity. The term is mostly used in complex analysis. Informally, it describes a function that has no singularities (like division by zero) for any finite complex number.
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning or usage. Spelling of related words follows regional conventions (e.g., 'holomorphic' vs. 'analytic' preference might vary slightly by textbook tradition, not geography).
Connotations
None beyond the technical mathematical meaning.
Frequency
Identically rare and specialized in both varieties, confined to university-level mathematics contexts.
Grammar
How to Use “entire function” in a Sentence
[The/An] [adjective] entire function + verb (e.g., grows, satisfies, has)Entire function of [order/type]Vocabulary
Collocations
Examples
Examples of “entire function” in a Sentence
adverb
British English
- The series converges entirely uniformly. (Note: 'entirely' is not directly derived from the technical term)
- The function behaves entirely differently. (General adverb)
American English
- The series converges entirely uniformly.
- The function behaves entirely differently.
adjective
British English
- The entire function property is crucial for the proof.
- We studied entire function theory.
American English
- The entire-function property is crucial for the proof.
- We studied entire-function theory.
Usage
Meaning in Context
Business
Never used.
Academic
Exclusively used in mathematics, specifically complex analysis. Found in textbooks, research papers, and lectures.
Everyday
Never used. Would cause confusion.
Technical
Core term in mathematical analysis and related engineering/physics fields (e.g., signal processing using complex analysis).
Vocabulary
Synonyms of “entire function”
Strong
Neutral
Weak
Vocabulary
Antonyms of “entire function”
Watch out
Common Mistakes When Using “entire function”
- Using 'entire' as a general adjective (e.g., 'the entire function was difficult' – ambiguous).
- Confusing with 'onto function' in set theory.
- Assuming it means a function whose range is all real numbers.
FAQ
Frequently Asked Questions
Yes, the complex sine function sin(z) is entire because its power series converges everywhere in the complex plane.
No, by definition, an entire function is holomorphic everywhere in the finite complex plane. A pole is a type of singularity, so it cannot be present.
All entire functions are holomorphic. 'Holomorphic' describes the property of complex differentiability on a domain. 'Entire' specifies that this domain is the entire complex plane.
Virtually never. It is a highly specialized term in complex analysis. In everyday or business English, the phrase 'entire function' would be parsed as the adjective 'entire' modifying the noun 'function' (e.g., 'the entire function was a disaster').
A complex-valued function that is holomorphic (complex-differentiable) at every point in the complex plane.
Entire function is usually formal, technical (exclusively academic mathematics) in register.
Entire function: in British English it is pronounced /ɪnˌtaɪə ˈfʌŋk.ʃən/, and in American English it is pronounced /ɪnˌtaɪər ˈfʌŋk.ʃən/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “As entire as they come (mathematical humour, implying no singularities)”
- “The entire caboodle (playful pun in technical writing)”
Learning
Memory Aids
Mnemonic
Think of the ENTIRE complex plane. If a function works nicely on the ENTIRE plane without breaking, it's an ENTIRE function.
Conceptual Metaphor
A perfectly smooth, unbreakable surface that extends infinitely in all directions (the complex plane).
Practice
Quiz
Which of the following is an example of an entire function?