mathematics

Identity theorem

Identity theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Identity theorem rather than just read about it. In short: In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ), if f = g on some S ⊆ D {\displaystyle S\subseteq D} , where S {\displaystyle S} has an accumulation point in D, then f = g on D. Thus an analytic function is…

Key takeaways

Identity theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Identity theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Identity theorem from memory before moving on to harder problems.

Reference excerpt

In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ), if f = g on some S ⊆ D {\displaystyle S\subseteq D} , where S {\displaystyle S} has an accumulation point in D, then f = g on D. Thus an analytic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D with an accumulation point (provided this contains a converging sequence together with its limit). This is not true in general for real-differentiable functions, even infinitely real-differentiable functions. In comparison, analytic functions are a much more rigid notion. The underpinning fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series. The connectedness assumption on the domain D is necessary. For example, if D consists of two disjoint open sets, f {\displaystyle f} can be 0 {\displaystyle 0} on one open set, and 1 {\displaystyle 1} on another, while g {\displaystyle g} is 0 {\displaystyle 0} on one, and 2 {\displaystyle 2} on another.

Lemma If two holomorphic functions f {\displaystyle f} and g {\displaystyle g} on a domain D agree on a set S which has an accumulation point c {\displaystyle c} in D {\displaystyle D} , then f = g {\displaystyle f=g} on a disk in D {\displaystyle D} centered at c {\displaystyle c} . To prove this, it is enough to show that f ( n ) ( c ) = g ( n ) ( c ) {\displaystyle f^{(n)}(c)=g^{(n)}(c)} for all n ≥ 0 {\displaystyle n\geq 0} , since both functions are analytic. If this is not the case, let m {\displaystyle m} be the smallest nonnegative integer with f ( m ) ( c ) ≠ g ( m ) ( c ) {\displaystyle f^{(m)}(c)\neq g^{(m)}(c)} . By holomorphy, we have the following Taylor series representation in some open neighborhood U of c {\displaystyle c} :

( f − g ) ( z )

= ( z − c ) m ⋅ [ ( f − g ) ( m ) ( c ) m ! + ( z − c ) ⋅ ( f − g ) ( m + 1 ) ( c ) ( m + 1 ) ! + ⋯ ]

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Identity theorem

Start with the simplest possible case. Write down what Identity theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Identity theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Identity theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Identity theorem

In research: Identity theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Identity theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Identity theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in complex analysis, Theorems in real analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Identity theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Identity theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Identity theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Identity theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Identity theorem in simple terms?

Why does Identity theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Identity theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Identity theorem.