Movable singularity

Key takeaways

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

Reference excerpt

In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in the sense that its location depends on the initial conditions of the differential equation. Suppose we have an ordinary differential equation in the complex domain. Any given solution y(x) of this equation may well have singularities at various points (i.e. points at which it is not a regular holomorphic function, such as branch points, essential singularities or poles). A singular point is said to be movable if its location depends on the particular solution we have chosen, rather than being fixed by the equation itself. For example the equation

d y d x = 1 2 y {\displaystyle {\frac {dy}{dx}}={\frac {1}{2y}}}

has solution y = x − c {\displaystyle y={\sqrt {x-c}}} for any constant c. This solution has a branchpoint at x = c {\displaystyle x=c} , and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant c). It is a basic feature of linear ordinary differential equations that singularities of solutions occur only at singularities of the equation, and so linear equations do not have movable singularities. When attempting to look for 'good' nonlinear differential equations it is this property of linear equations that one would like to see: asking for no movable singularities is often too stringent, instead one often asks for the so-called Painlevé property: 'any movable singularity should be a pole', first used by Sofia Kovalevskaya.

See also Painlevé transcendents Regular singular point

References

Einar Hille (1997), Ordinary Differential Equations in the Complex Domain, Dover. ISBN 0-486-69620-0

Illustrations

Worked examples

Example 1 — a first encounter with Movable singularity

Start with the simplest possible case. Write down what Movable singularity 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 Movable singularity 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 Movable singularity 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 Movable singularity

In research: Movable singularity 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 Movable singularity 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: Movable singularity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Complex analysis, Ordinary differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Movable singularity 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.

How to study Movable singularity in 20 minutes

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

Frequently asked questions

What is Movable singularity in simple terms?

In the theory of ordinary differential equations, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in the sense that its location depends on the initial conditions of the differential equation. Suppose we have an ordinary differential equation…

Why does Movable singularity 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 Movable singularity?

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 Movable singularity.