mathematics

Lommel function

Lommel function 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 Lommel function rather than just read about it. In short: The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation: z 2 d 2 y d z 2 + z d y d z + ( z 2 − ν 2 ) y = z μ + 1 . {\displaystyle z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.} Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by Eugen von Lommel (1880), s μ , ν ( z ) = π 2 [ Y ν ( z ) ∫ 0 z…

Key takeaways

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

Reference excerpt

The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:

z 2 d 2 y d z 2 + z d y d z + ( z 2 − ν 2 ) y = z μ + 1 . {\displaystyle z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.}

Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by Eugen von Lommel (1880),

s μ , ν ( z ) = π 2 [ Y ν ( z ) ∫ 0 z x μ J ν ( x ) d x − J ν ( z ) ∫ 0 z x μ Y ν ( x ) d x ] , {\displaystyle s_{\mu ,\nu }(z)={\frac {\pi }{2}}\left[Y_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }J_{\nu }(x)\,dx-J_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }Y_{\nu }(x)\,dx\right],}

S μ , ν ( z ) = s μ , ν ( z ) + 2 μ − 1 Γ ( μ + ν + 1 2 ) Γ ( μ − ν + 1 2 ) ( sin ⁡ [ ( μ − ν ) π 2 ] J ν ( z ) − cos ⁡ [ ( μ − ν ) π 2 ] Y ν ( z ) ) , {\displaystyle S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}\Gamma \left({\frac {\mu +\nu +1}{2}}\right)\Gamma \left({\frac {\mu -\nu +1}{2}}\right)\left(\sin \left[(\mu -\nu ){\frac {\pi }{2}}\right]J_{\nu }(z)-\cos \left[(\mu -\nu ){\frac {\pi }{2}}\right]Y_{\nu }(z)\right),}

where Jν(z) is a Bessel function of the first kind and Yν(z) a Bessel function of the second kind. The s function can also be written as

s μ , ν ( z ) = z μ + 1 ( μ − ν + 1 ) ( μ + ν + 1 )

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Lommel function

Start with the simplest possible case. Write down what Lommel function 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 Lommel function 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 Lommel function 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 Lommel function

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

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

Frequently asked questions

What is Lommel function in simple terms?

The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation: z 2 d 2 y d z 2 + z d y d z + ( z 2 − ν 2 ) y = z μ + 1 . {\displaystyle z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.} Solutions are given b…

Why does Lommel function 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 Lommel function?

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 Lommel function.