Radial function

Key takeaways

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

Reference excerpt

In mathematics, a radial function is a real-valued function defined on a Euclidean space ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance. For example, a radial function Φ in two dimensions has the form

Φ ( x , y ) = φ ( r ) , r = x 2 + y 2 {\displaystyle \Phi (x,y)=\varphi (r),\quad r={\sqrt {x^{2}+y^{2}}}}

where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical functions, and any descent function (e.g., continuous and rapidly decreasing) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the solid spherical harmonic expansion. A function is radial if and only if it is invariant under all rotations leaving the origin fixed. That is, f is radial if and only if

f ∘ ρ = f {\displaystyle f\circ \rho =f\,}

for all ρ ∈ SO(n), the special orthogonal group in n dimensions. This characterization of radial functions makes it possible also to define radial distributions. These are distributions S on ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ such that

S [ φ ] = S [ φ ∘ ρ ] {\displaystyle S[\varphi ]=S[\varphi \circ \rho ]}

for every test function φ and rotation ρ. Given any (locally integrable) function f, its radial part is given by averaging over spheres centered at the origin. To wit,

ϕ ( x ) = 1 ω n − 1 ∫ S n − 1 f ( r x ′ ) d x ′ {\displaystyle \phi (x)={\frac {1}{\omega _{n-1}}}\int _{S^{n-1}}f(rx')\,dx'}

where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and r = |x|, x′ = x/r. It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every r. The Fourier transform of a radial function is also radial, and so radial functions play a vital role in Fourier analysis. Furthermore, the Fourier transform of a radial function typically has stronger decay behavior at infinity than non-radial functions: for radial functions bounded in a neighborhood of the origin, the Fourier transform decays faster than R−(n−1)/2. The Bessel functions are a special class of radial function that arise naturally in Fourier analysis as the radial eigenfunctions of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform.

Worked examples

Example 1 — a first encounter with Radial function

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

In research: Radial 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 Radial 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: Radial function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Harmonic analysis, Rotational symmetry, Types of functions, so understanding it makes those chapters shorter.
In everyday life: Look for Radial 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.

How to study Radial function in 20 minutes

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

Frequently asked questions

What is Radial function in simple terms?

In mathematics, a radial function is a real-valued function defined on a Euclidean space ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance.

Why does Radial 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 Radial 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 Radial function.