mathematics

Trigamma function

Trigamma 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 Trigamma function rather than just read about it. In short: In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln ⁡ Γ ( z ) {\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)} . It follows from this definition that ψ 1 ( z ) = d d z ψ ( z ) {\displaystyle \psi _{1}(z)={\frac {d}{dz}}\psi (z)} where ψ(z) is the digamma function.

Key takeaways

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

Reference excerpt

In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by

ψ 1 ( z ) = d 2 d z 2 ln ⁡ Γ ( z ) {\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)} . It follows from this definition that

ψ 1 ( z ) = d d z ψ ( z ) {\displaystyle \psi _{1}(z)={\frac {d}{dz}}\psi (z)}

where ψ(z) is the digamma function. It may also be defined as the sum of the series

ψ 1 ( z ) = ∑ n = 0 ∞ 1 ( z + n ) 2 , {\displaystyle \psi _{1}(z)=\sum _{n=0}^{\infty }{\frac {1}{(z+n)^{2}}},}

making it a special case of the Hurwitz zeta function

ψ 1 ( z ) = ζ ( 2 , z ) . {\displaystyle \psi _{1}(z)=\zeta (2,z).}

Note that the last two formulas are valid when 1 − z is not a natural number.

Calculation A double integral representation, as an alternative to the ones given above, may be derived from the series representation:

ψ 1 ( z ) = ∫ 0 1 ∫ 0 x x z − 1 y ( 1 − x ) d y d x {\displaystyle \psi _{1}(z)=\int _{0}^{1}\!\!\int _{0}^{x}{\frac {x^{z-1}}{y(1-x)}}\,dy\,dx}

using the formula for the sum of a geometric series. Integration over y yields:

ψ 1 ( z ) = − ∫ 0 1 x z − 1 ln ⁡ x 1 − x d x {\displaystyle \psi _{1}(z)=-\int _{0}^{1}{\frac {x^{z-1}\ln {x}}{1-x}}\,dx}

An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Trigamma function

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

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

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

Frequently asked questions

What is Trigamma function in simple terms?

In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln ⁡ Γ ( z ) {\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)} . It follows from this definition that ψ 1 ( z ) = d d z ψ ( z ) {\displ…

Why does Trigamma 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 Trigamma 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 Trigamma function.