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I am writing some Calculus content, and I would like a "big list" of useful functions which are defined by definite integrals, but are not elementary functions.

Two examples of such functions are

$$ \mathrm{Erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\mathrm{d} t $$

which is fundamentally important to statistics, and

$$ \mathrm{Si}(x) = \int_0^x \frac{\sin(t)}{t} \mathrm{d} t $$

which comes up all the time in signal processing.

I would like to be able to sketch such functions, express some definite integrals (like $\int_0^1 e^{-4t^2} dt$) in terms of such functions, etc.

So what other functions are important enough to have their own name, and are given as integrals of elementary functions?

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    $\begingroup$ I am very surprised that no one has mentioned the NIST Digital Library of Mathematical Functions, which has a very thorough collection of integral representations (among many, many other things) for almost any special function you could name. $\endgroup$ – Robert Mastragostino Dec 3 '15 at 16:08
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    $\begingroup$ @RobertMastragostino Awesome resource! $\endgroup$ – Steven Gubkin Dec 3 '15 at 17:47
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    $\begingroup$ Another collection of resources about elementary integration can be found in Andrés Caicedo's answer to MSE 287442. $\endgroup$ – Benjamin Dickman Dec 3 '15 at 20:52
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It seems that the key term here may be the somewhat non-specific-sounding special functions.

By googling for a few examples (Erf, Si, Li) I came across a Table of Special Functions and, on the Lists of integrals wikipage, there is a sub-section on Special Functions.

As a related remark, one reason that functions may be presented and/or defined in terms of integrals is that it may not be possible to express them using elementary functions. Actually proving that an antiderivative cannot be written using elementary functions is generally done using Liouville's criterion; in the spirit of the OP's mention of $\text{Si}(x)$, I include here a pointer to MSE 694915 in which I provided a reference with a proof that the antiderivative of $\sin(x)/x$ cannot be expressed using elementary functions alone.

For one other example, see Peter Mueller's proof that the antiderivative of $x \tan x$ cannot be written using elementary functions in MO 108598 (for an additional reference proving this fact, see the answer that I provided at the same question). Although I am not aware of an important function defined directly in terms of an integral of $x \tan x$, one can use the relation between its antiderivative and the dilogarithm to show that the latter, too, cannot be expressed in terms of elementary functions.

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    $\begingroup$ great answer, i thought the related info was appropriate and helpful $\endgroup$ – celeriko Dec 2 '15 at 23:25
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    $\begingroup$ Thanks, this is great. I should probably get around to actually understanding the proof of Liouville's criterion one day, since I always mention this is class... $\endgroup$ – Steven Gubkin Dec 2 '15 at 23:38
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    $\begingroup$ @StevenGubkin For a powerhouse of an application of this criterion, see (or recall) Robert Bryant's use of it in MO 171733... $\endgroup$ – Benjamin Dickman Dec 2 '15 at 23:56
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The gamma function is very useful in counting problems (among others) and is seen as an extension of the factorial function into the reals. It is defined as:

$$ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,. $$

(Incidentally, this is the example of how to use MathJax in the help section.)

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My first take is $$ \ln(x) = \int_1^x\frac1t dt. $$ Granted, some texts introduce the natural log of the inverse of $\exp$ but other texts define $\ln$ as above and the $\exp$ as the inverse. If I remember correctly, the definition of the logarithm by the integral was historically first.

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  • $\begingroup$ I think this answer fits the spirit of the question, but isn't $\ln(x)$ generally considered among the list of "elementary" functions? $\endgroup$ – Michael Joyce Dec 3 '15 at 14:29
  • $\begingroup$ This is great as well, and well worth mentioning. But it is in my list of "elementary functions". I would find it surprising if the integral definition came first. I would imagine the first interpretation of $\ln(x)$ was probably something along the lines of " I make a table of $1.001^n$. If $1.001^n = x$, then I define $\ln x = \frac{n}{1000}$". This would arise naturally in attempts to solve exponential equations. Maybe not exactly this, but some history involving exponential equation solving first. Just a guess though. $\endgroup$ – Steven Gubkin Dec 3 '15 at 14:35
  • $\begingroup$ Article on the history of logarithms here: en.wikipedia.org/wiki/History_of_logarithms $\endgroup$ – Daniel R. Collins Dec 3 '15 at 15:59
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    $\begingroup$ In this sense you can add $\arctan(x) = \int_0^x (1+t^2)^{-1/2}\,dt$. $\endgroup$ – Gerald Edgar Dec 3 '15 at 16:05
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    $\begingroup$ Actually, my own analysis course (in Germany) went like this: Once we had integrals, we defined $\ln$ as the antiderivative of $1/x$, then we showed some properties e.g. that $\ln:]0,\infty[\to\mathbb{R}$ stricty increasing and onto, hence invertible, called the inverse $\exp$ and derived the power series for $\exp$. So this is a pretty natural example for me. I also found calculus lecture notes from the US that take a similar route. @GeraldEdgar Do you know a text that introduces $\arctan$ as an integral and defines $\tan$ as inverse of that? $\endgroup$ – Dirk Dec 3 '15 at 19:33
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The function $\displaystyle \text{Li}(x) = \int_2^x \frac{1}{\log t} \,\, dt$ comes up in the study of the distribution of primes. Specifically, the number of prime numbers less than $x$ is asymptotic to $\text{Li}(x)$ and a major consequence of the Riemann Hypothesis would be the sharpest possible bound for the difference between these two functions as $x \rightarrow \infty$.

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No one has still mentioned Fresnel functions:

$S(x)=\int_0^x \sin(t^2)dt$ and $C(x)= \int_0^x \cos(t^2)dt$

They are (of course) very relevant in signal analysis and in studying diffraction. What is less often mentioned is that the parametric function $x\mapsto (S(x),C(x))$ gives you a beautiful curve, the Cornu spiral, which is used by engineers in roads and railorads design and which is a very nice curve from the differential geometric point of view since its curvature is a linear function.

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    $\begingroup$ Why do engineers interested in (rail)road design care about the Cornu spiral? $\endgroup$ – KCd Dec 6 '15 at 5:24
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    $\begingroup$ Since curvature is varying linearly the best way to design a curve is as an arc of Cornu spiral, If curves in railroads were designed as arc of circles there would be a point of sharp variation of curvature from 0 to 1/R (curvature would not be continuous). Since curvature is proportional to centrifugal force this would mean an impulsive force acting on the vehicle arcs of Cornu spirals allows to design connections between different parts of roads with overall continuous curvature. $\endgroup$ – Nicola Ciccoli Dec 7 '15 at 12:13
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A (cata)caustic is formed by the reflection of light, such as the cardioid in this coffee cup:

G.B. Airy showed in [Airy, "On the intensity of light in the neighbourhood of a caustic," Transactions of the Cambridge Philosophical Society 6 (1838) 379--402] that the intensity of the light along a normal line to the caustic is proportional to the square of the integral $$ \int_0^{\infty } \cos \frac{\pi}{2} \left(w^3-m w\right) \; dw $$ where $m$ is proportional to the displacement from the caustic toward the convex side. This led to the Airy functions and the Airy differential equation. The modern definition is $$ \mathop{\mathrm{Ai}}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\cos}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt $$ and the differential equation is $y''=x\,y$. A second independent solution to the differential equation $Bi(x)$ also has an integral representation, but it is not so nice.

As a side note, if your calculus or differential equations courses teach students how to think with factorial-like products that multiply every other integer (e.g. the power series of Bessel functions, which also have an integral representation) or as in this case, multiply every third, Airy's example shows a natural application of such formulas. The Airy functions are linear combinations of the series $$ y_1 = 1+\frac{1}{3!}x^{3}+\frac{1\cdot 4}{6!}x^{6}+% \frac{1\cdot 4\cdot 7}{9!}x^{9}+\cdots $$ $$ y_2=x+\frac{2}{4!}x^{4}+\frac{2\cdot 5}{7!}x^{7}+\frac{2% \cdot 5\cdot 8}{10!}x^{10}+\cdots $$

Remark: While the Airy (and Bessel) functions are (or can be) defined by definite integrals, they are "complete" integrals (i.e., with fixed limits of integration) that are functions of a parameter in the integrand, whereas the OP's examples have one limit of integration that varies. The gamma function is also of the first class (i.e. "complete"). The significance is that the OP's class of examples can be used in single-variable calculus. The other class requires differentiating under the integral sign, which is usually not taught until multivariable calculus or later.

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    $\begingroup$ +1 and I wonder if this could make a good answer at MESE 7528 (Real-world examples of more “obscure” geometric figures), too. $\endgroup$ – Benjamin Dickman Mar 28 '16 at 5:32
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Mathematica has many special functions built into its language. There's a whole list here:

https://reference.wolfram.com/language/guide/SpecialFunctions.html

Many of these are defined from definite integrals. You can find the definitions in the details section of a specific function.

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