mathematics

Picard theorem

Picard 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 Picard theorem rather than just read about it. In short: In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

Key takeaways

Picard 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 Picard theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Picard theorem from memory before moving on to harder problems.

Reference excerpt

In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

The theorems

Little Picard Theorem: If a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is entire and non-constant, then the set of values that f ( z ) {\textstyle f(z)} assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by λ {\textstyle \lambda } , and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If f {\textstyle f} omits two values, then lifting f {\textstyle f} along the universal covering map sends the plane into the unit disc via a holomorphic function, which implies that f {\textstyle f} is constant by Liouville's theorem. This theorem is a significant strengthening of Liouville's theorem which states that the image of an entire non-constant function must be unbounded. Many different proofs of Picard's theorem were later found and Schottky's theorem is a quantitative version of it. In the case where the values of f {\textstyle f} are missing a single point, this point is called a lacunary value of the function.

Great Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity at a point w {\textstyle w} , then on any punctured neighborhood of w , f ( z ) {\textstyle w,f(z)} takes on all possible complex values, with at most a single exception, infinitely often. This is a substantial strengthening of the Casorati–Weierstrass theorem, which only guarantees that the range of f {\textstyle f} is dense in the complex plane. A result of the Great Picard Theorem is that any entire, non-polynomial function attains all possible complex values infinitely often, with at most one exception. The "single exception" is needed in both theorems, as demonstrated here:

ez is an entire non-constant function that is never 0,

e 1 z {\textstyle e^{\frac {1}{z}}} has an essential singularity at 0, but still never attains 0 as a value.

Proof

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Picard theorem

Start with the simplest possible case. Write down what Picard 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 Picard 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 Picard 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 Picard theorem

In research: Picard 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 Picard 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: Picard theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Picard 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.

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How to study Picard theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Picard 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 Picard theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Picard theorem in simple terms?

In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

Why does Picard 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 Picard 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 Picard theorem.