mathematics

Sonine formula

Sonine formula 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 Sonine formula rather than just read about it. In short: In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin. One such formula is the following integral formula involving a product of three Bessel functions: ∫ 0 ∞ J z ( a t ) J z ( b t ) J z ( c t ) t 1 − z d t = 2 z − 1 Δ ( a , b , c ) 2 z − 1 π 1 / 2 Γ ( z + 1 2 ) ( a b c ) z {\displaystyle \int _{0}^{\infty }J_{z}(at)J_{z}(bt)J_{z}(ct)t^{1-z}\,dt={\f…

Key takeaways

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

Reference excerpt

In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin. One such formula is the following integral formula involving a product of three Bessel functions:

∫ 0 ∞ J z ( a t ) J z ( b t ) J z ( c t ) t 1 − z d t = 2 z − 1 Δ ( a , b , c ) 2 z − 1 π 1 / 2 Γ ( z + 1 2 ) ( a b c ) z {\displaystyle \int _{0}^{\infty }J_{z}(at)J_{z}(bt)J_{z}(ct)t^{1-z}\,dt={\frac {2^{z-1}\Delta (a,b,c)^{2z-1}}{\pi ^{1/2}\Gamma (z+{\tfrac {1}{2}})(abc)^{z}}}}

where Δ is the area of a triangle with given sides.

References Stempak, Krzysztof (1988), "A new proof of Sonine's formula", Proceedings of the American Mathematical Society, 104 (2): 453–457, doi:10.2307/2046994, ISSN 0002-9939, JSTOR 2046994, MR 0962812

Worked examples

Example 1 — a first encounter with Sonine formula

Start with the simplest possible case. Write down what Sonine formula 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 Sonine formula 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 Sonine formula 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 Sonine formula

In research: Sonine formula 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 Sonine formula 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: Sonine formula is common in secondary-school and first-year university syllabi. It links to neighbouring topics Special hypergeometric functions, so understanding it makes those chapters shorter.
In everyday life: Look for Sonine formula 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 Sonine formula in 20 minutes

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

Frequently asked questions

What is Sonine formula in simple terms?

Why does Sonine formula 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 Sonine formula?

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 Sonine formula.