Digital Library of Mathematical Functions
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Guide to Searching the DLMF

§ Overview

In DLMF, you can search not only for text but also for mathematical expressions. Every page contains a search box in the navigation bar. From there you can also access an advanced search page where you can control certain settings, narrowing the search to certain chapters, or restricting the results to equations, graphs, tables, or bibliographic items.

Table 1: Query Examples
Query Matching records contain
"Fourier transform" and series both the phrase “Fourier transform” and the word “series”.
Fourier or series at least one of the words “Fourier” or “series”.
Fourier (transform or series) at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform" both 1/(2π) and the phrase “Fourier transform”.
sin^2 +cos^2 the expression sin2+cos2.
J_n(z)= the math fragment Jn(z)=
J_n@(z)= the math fragment Jn(z)=, emphasizing more that Jn is a function.
J_n@(x or z)= at least one of the math fragments Jn(x)= or Jn(z), emphasizing that Jn is a function.
int sin the integral of the sin function
int_$^$ sin any definite integral of sin
sin x and (J_nu(z) or I_nu(z)) both sinx and at least one of the two functions Jν(z) or Iν(z).
DeMoivre and cos (n theta) both the word “DeMoivre” and the expression cos(nθ).
BesselJ_nu and BesselY_nu both the Bessel functions Jν and Yν.
Euler the word ”Euler” or any of the various Euler terms such as Euler Gamma function Γ, Euler Beta function B, etc.
trigonometric the word ”trigonometric” or any of the various trigonometric functions such as sin, cos, tan, and cot.
trig$ any word that matches the pattern “trig$”, that is, that starts with “trig”, such as “trigonometric” and “trigonometry”, including the various trigonometric functions such as sin and cos.
trigonometric^2 + trig$^2 any sum of the squares of two trigonometric functions such as sin2z+cos2z.
int adj sin immediately followed by sin without any intervening terms.
int prec/10 sin(x) preceding sinx by no more than 10 terms.
Gamma near/5 = the terms Γ and = such that Γ is up to 5 terms before or after =.
(int adj sin(x)) near/5 = the sub-expression sin and =, in whatever order and separated by no more than 5 terms.

§ More Details

§ Terms, Phrases and Expressions

Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators:

term:

a textual word, a number, or a math symbol.

phrase:

any double-quoted sequence of textual words and numbers.

math expression:

any LaTeX-like sequence of terms, excluding phrases.

Boolean operator:

and and or

proximity operator:

adj, prec/n, and near/n, where n is any positive natural number.

All terms are taken to be case-insensitive, except those taken to represent math expressions (see Case Sensitivity).

§ Wildcards

Wildcards allow matching patterns and marking parts of an expression that don’t matter (as for example, which variable name the author uses for a function):

$ stands for any number of alphanumeric characters
(the more conventional * is reserved for the multiplication operator)
? stands for any zero or one alphanumeric character.
(If ? occurs as a separate token, it effectively acts like $)
Table 2: Wildcard Examples
Query What it stands for
sin? sin, sine, sinh, sinc, …
si$ si, sin, sine, sinh, sinc, sinInt, similar, …
co$te Coordinate, conjugate, correlate,…
B$l? Bessel, BesselJ, BesselI, BesselK, Bernoulli,…
int_$^$ sin any definite integral of sin.

§ Query relaxation

If a query does not return any hits, the DLMF search system relaxes the query to match and retrieve what may be intended by the original query. For example, the expression Ai2+Bi2 does not occur verbatim in DLMF, but Ai2(z)+Bi2(z) and Ai2(x)+Bi2(x) do. Therefore, if your query is Ai^2+Bi^2, the system modifies the query so it will find the equations containing the latter expressions.

§ Fine Points in Math Search

To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. Be careful, however, because if you put quotes around math expressions involving math symbols, you may not get the matches you’d expect.

§ Math Symbols

You can use in math queries all the symbols and commands defined in LaTeX (you can omit the \), and some additional convenient ones, as well as the special functions’ names:

Table 3: A sample of recognized symbols
Symbols Comments
=, -, +, *, /, >, <
<=, >= For and .
+-, -+ For ± and
_, ^ For subscripting and superscripting (power)
' (single quote) As a prime, which can stand for the derivative
->, <-, <->, =>, <==, <=> For arrows , , , , and
~ For similar-to
~~ For approximation .
-= For equivalence
All elementary functions Such as sin, cos, tan, Ln, log, exp
All names of special functions Such as BesselI, BesselJ, LegendreP, KummerU, etc.

The syntax of the special functions can be LaTeX-like or as employed in widely used computer algebra systems. For example, for the Bessel function Kn(z), you can write K_n(z), BesselK_n(z), BesselK(n,z), or BesselK[n,z]. Note that the first form may match other functions K than the Bessel K function, so if you are sure you want Bessel K, you might as well enter one of the other 3 forms.

§ Case Sensitivity

DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. In the following situations, DLMF search is case-sensitive:

  • Single-letter terms

  • All the Greek Letters (gamma vs. Gamma, sigma vs. Sigma, etc.)

  • All the inverse trigonometric functions (arcsin vs. Arcsin, etc.).

  • The following standard special functions: si, Si, ci, Ci, shi, Shi, ce, Ce, se, Se, ln, Ln, Lommels, LommelS, Jacobiphi, JacobiPhi, and the list is still growing.

§ Fonts and Accents

Sometimes there are distinctions between various special function names based on font style, such as the use of bold or calligraphic letters. DLMF search recognizes just the essential font differences, that is, the font style differences deemed important for the DLMF contents:

Table 4: Font and Accent Examples
Query Matches
cal L
bold H H
theta hat θ^
U bar U¯
B tilde B~

If you don’t specify the font style or font accessories in the query, the style and accessories won’t matter in the search, but if you specify them, they will matter.

§ Metadata for More Effective Search

To find more effectively the information you need, especially equations, you may at times wish to specify what you want with descriptive words that characterize the contents but do not occur literally. For example, you may want equations that contain trigonometric functions, but you don’t care which trigonometric function. The natural thing is to enter the query trigonometric, and expect to have all the right matches. DLMF search provides this feature, which it accomplishes through the use of metadata.