mathematics

Inverse tangent integral

Inverse tangent integral 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 Inverse tangent integral rather than just read about it. In short: The inverse tangent integral is a special function, defined by: Ti 2 ⁡ ( x ) = ∫ 0 x arctan ⁡ t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt} Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function. Definition The inverse tangent integral is defined by: Ti 2 ⁡ ( x ) = ∫ 0 x arctan ⁡ t t d t {\displaystyle \operat…

Key takeaways

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

Reference excerpt

The inverse tangent integral is a special function, defined by:

Ti 2 ⁡ ( x ) = ∫ 0 x arctan ⁡ t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt}

Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function.

Definition The inverse tangent integral is defined by:

Ti 2 ⁡ ( x ) = ∫ 0 x arctan ⁡ t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt}

The arctangent is taken to be the principal branch; that is, −π/2 < arctan(t) < π/2 for all real t. Its power series representation is

Ti 2 ⁡ ( x ) = x − x 3 3 2 + x 5 5 2 − x 7 7 2 + ⋯ {\displaystyle \operatorname {Ti} _{2}(x)=x-{\frac {x^{3}}{3^{2}}}+{\frac {x^{5}}{5^{2}}}-{\frac {x^{7}}{7^{2}}}+\cdots }

which is absolutely convergent for | x | ≤ 1. {\displaystyle |x|\leq 1.}

The inverse tangent integral is closely related to the dilogarithm Li 2 ⁡ ( z ) = ∑ n = 1 ∞ z n n 2 {\textstyle \operatorname {Li} _{2}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{2}}}} and can be expressed simply in terms of it:

Ti 2 ⁡ ( z ) = 1 2 i ( Li 2 ⁡ ( i z ) − Li 2 ⁡ ( − i z ) ) {\displaystyle \operatorname {Ti} _{2}(z)={\frac {1}{2i}}\left(\operatorname {Li} _{2}(iz)-\operatorname {Li} _{2}(-iz)\right)}

That is,

Ti 2 ⁡ ( x ) = Im ⁡ ( Li 2 ⁡ ( i x ) ) {\displaystyle \operatorname {Ti} _{2}(x)=\operatorname {Im} (\operatorname {Li} _{2}(ix))}

for all real x.

Properties The inverse tangent integral is an odd function:

Ti 2 ⁡ ( − x ) = − Ti 2 ⁡ ( x ) {\displaystyle \operatorname {Ti} _{2}(-x)=-\operatorname {Ti} _{2}(x)}

The values of Ti2(x) and Ti2(1/x) are related by the identity

Ti 2 ⁡ ( x ) − Ti 2 ⁡ ( 1 x ) = π 2 log ⁡ x {\displaystyle \operatorname {Ti} _{2}(x)-\operatorname {Ti} _{2}\left({\frac {1}{x}}\right)={\frac {\pi }{2}}\log x}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Inverse tangent integral

Start with the simplest possible case. Write down what Inverse tangent integral 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 Inverse tangent integral 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 Inverse tangent integral 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 Inverse tangent integral

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

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

Frequently asked questions

What is Inverse tangent integral in simple terms?

The inverse tangent integral is a special function, defined by: Ti 2 ⁡ ( x ) = ∫ 0 x arctan ⁡ t t d t {\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt} Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special funct…

Why does Inverse tangent integral 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 Inverse tangent integral?

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 Inverse tangent integral.