cauchy integral formula: meaning, definition, pronunciation and examples
C2 (Highly specialized technical term)Formal academic / Technical mathematical
Quick answer
What does “cauchy integral formula” mean?
A central theorem in complex analysis stating that the value of a holomorphic function at a point inside a simple closed contour can be expressed as a contour integral involving the function's values on the contour.
Audio
Pronunciation
Definition
Meaning and Definition
A central theorem in complex analysis stating that the value of a holomorphic function at a point inside a simple closed contour can be expressed as a contour integral involving the function's values on the contour.
In mathematics, a fundamental result connecting local behavior of analytic functions to their boundary values, widely used in solving boundary value problems, evaluating integrals, and proving other theorems like Taylor series expansions.
Dialectal Variation
British vs American Usage
Differences
No lexical differences; identical technical usage. British texts may use 'formulae' as plural more frequently than American 'formulas'.
Connotations
Identical connotations of mathematical rigor and foundational complex analysis.
Frequency
Equally rare outside specialized mathematics; slightly higher frequency in British texts on pure mathematics.
Grammar
How to Use “cauchy integral formula” in a Sentence
[Theorem/Formula] + [verb: provides/yields/gives] + [result][One] + [verb: applies/uses] + [Cauchy integral formula] + [preposition: to/for] + [purpose]Vocabulary
Collocations
Examples
Examples of “cauchy integral formula” in a Sentence
verb
British English
- One cauchies the integral to evaluate the series.
- The function was cauchied around the singularity.
American English
- We Cauchy the formula to solve the boundary problem.
- He Cauchied the integral representation.
adverb
British English
- The function behaves Cauchy-integrally near the pole.
- It was derived Cauchy-integral-wise.
American English
- The solution was found Cauchy-integral-style.
- Proceed Cauchy-formula-ly.
adjective
British English
- The Cauchy-integral approach proved most efficient.
- This is a Cauchy-type representation.
American English
- The Cauchy-integral method is standard.
- We need a Cauchy-formula solution.
Usage
Meaning in Context
Business
Not used
Academic
Core concept in graduate-level complex analysis, used in proofs and problem sets.
Everyday
Never used
Technical
Essential in mathematical physics, fluid dynamics, electromagnetism, and engineering analysis.
Vocabulary
Synonyms of “cauchy integral formula”
Strong
Neutral
Weak
Vocabulary
Antonyms of “cauchy integral formula”
Watch out
Common Mistakes When Using “cauchy integral formula”
- Misspelling as 'Couchy' or 'Cauchy's integral theorem' (related but distinct)
- Applying to non-holomorphic functions
- Forgetting the 1/(2πi) factor.
FAQ
Frequently Asked Questions
No. Cauchy's theorem states the integral of a holomorphic function over a closed contour is zero. The integral formula provides a specific representation for the function's values inside the contour.
No, only to functions that are holomorphic (analytic) inside and on a simple closed contour.
It comes from the residue of the simple pole at the point of evaluation and ensures the formula yields the correct function value.
Pure mathematics (complex analysis), theoretical physics, electrical engineering (signal processing), and fluid dynamics.
A central theorem in complex analysis stating that the value of a holomorphic function at a point inside a simple closed contour can be expressed as a contour integral involving the function's values on the contour.
Cauchy integral formula is usually formal academic / technical mathematical in register.
Cauchy integral formula: in British English it is pronounced /ˈkəʊʃi ˈɪntɪɡrəl ˈfɔːmjʊlə/, and in American English it is pronounced /ˈkoʊʃi ˈɪntəɡrəl ˈfɔrmjələ/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “As fundamental as Cauchy's formula”
- “Cauchy's formula in action”
Learning
Memory Aids
Mnemonic
Cauchy Captures Inside Values using Outside Contours: CIVOC
Conceptual Metaphor
A mathematical crystal ball that predicts a function's behavior everywhere inside a region by looking only at its boundary.
Practice
Quiz
What is a primary application of the Cauchy integral formula?