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I am an undergrad in math and was wondering, what are for you the most important functions every mathematician should know? At the moment I think sin(x), cos(x), tan(x) ect. are extremely important for me.

PS.: I am sorry if my question is against the usual format here on math.stackexchange. However, I am really curious about your opinion on this topic. That`s the reason why I am asking.

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closed as primarily opinion-based by G.T.R, Belgi, Hakim, Pece, Henning Makholm Jun 27 at 17:31

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

3  
Polynomial functions? –  Seirios Jun 27 at 8:54
4  
The identity function should definitely be on that list. –  Cameron Buie Jun 27 at 8:55
3  
Trigonometric functions should be known to anyone, not just mathematicians. –  O.L. Jun 27 at 8:56
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I'd say if it has a name, it's probably important enough to know for at least most mathematicians. –  Silynn Jun 27 at 8:58
    
I see the question has not been declared CW. Presumably this means we expect one definitive answer. –  GEdgar Jun 27 at 14:28

7 Answers 7

up vote 8 down vote accepted

Some functions that have counter-intuitive properties on $\mathbb R$ that you should know :

  • The Weierstrass function, continuous everywhere, differentiable nowhere function.
  • The Cantor function, a uniformly continuous function that is not absolutely continuous.
  • The Minkowski's question mark function, an increasing continuous function, non differentiable on rational numbers only, and bijective from dyadic numbers to rational numbers and from rational numbers to algebraic numbers of degree 2.
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+1 for the ?(x) function, this one is new to me! –  Tyler Jun 27 at 16:31

As I see it, the important functions that every mathematician should know are:

  • $\Gamma(s)$
  • $\zeta(s)$
  • $sinh(x)$
  • $ch(x)$
  • $ln(x)$

(Where $s$ is a complex variable, and $x$ is a real number.)

Because they are important in pure mathematics and important in number theory, and we know that almost all unsolved mathematics are focused in number theory.

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2  
So, for example, $\exp(x)$ is unimportant... –  Ruslan Jun 27 at 13:05
    
What is $sh$? I assume it is not $\sinh$, since you also write $\cosh$ and not $ch$. –  GEdgar Jun 27 at 14:26
    
$\cosh$ stands for the hyperbolic cosine. –  Gahawar Jun 27 at 14:59
    
ok, thanks, now it's fixed –  rafik Jun 27 at 15:02

Any function with "canonical" in the name. And the identity function. (I've been getting perversely much mileage out of the identity function lately.)

  • canonical projection takes an element to its equivalence class (really set theory, but usually first seen in group theory)
  • canonical projection morphism takes an element of a product to a specific one of its factors (really set/category theory, but usually first seen in topology, or earlier, e.g., "$x$-component")
  • canonical transformation (see symplectomorphism), especially relevant in the Lie symmetry solution of differential equations

There are more such. Related, but not functions: canonical model, canonical singularity. (In fact, if it's "canonical", you should know about it.)

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This also the synonym(ish) natural (for the categorically inclined ;p ). –  Moses Jun 28 at 7:32

I always though the exponential function (and its generalizations) was widely regarded as the most mathematically significant function. From a complex analysis point of view, the trig. functions all come under the umbrella of the exponential function. Also, the exponential map is central in many areas closely connected to differential geometry, there's the matrix exponential, there's exponentiation of an operator that's used in functional analysis and quantum mechanics, etc.

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1) Polynomial functions

2) rational functions

3) trigonometric functions

4) hyperbolic functions

5) $e^x$ (also the complex e-function) and $log_a(b)$

And also many special functions like the dirichlet function, because its not riemann-integrable, but lebesgue integrable. (Nowhere continous), the squared function, which is not lipschitz-continous, but uniformly continuous! And maybe its useful to know all the (anti)-integrals of them for example: $(arctan(x))'=\frac{1}{x^2+1}$

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I dont think there is a list of functions that "every mathematician should know". There are simply to many functions, and besides the elementary functions (polynomials, rational functions, log, exp, power functions, trig and inverse trig functions) one must learn what is needed. In statistics, for instance, we use a lot special functions such as the gamma function, the beta function, others). There is a huge number of so-called special functions, which one usually must learn as need arises. You can search this site for the special-function tag (which I just added to your post), or have a look at this online catalog: http://dlmf.nist.gov/ (NIST Digital Library of Mathematical Functions)

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Please explain why did you downvote my answer. Though seeing the result, I already have an idea. By the way one of your upvotes is mine, clever guy. –  O.L. Jun 27 at 15:22
    
The reason you give in unconvincing, especially because before the downvote both answers had equal score. –  O.L. Jun 27 at 15:31
    
well, what can I do? If you look at my personal page you will see I am a quite heavy downvoter, with a total of 65 downvotes cast. –  kjetil b halvorsen Jun 27 at 16:02
    
sure, sure. And I suppose in all 65 cases you waited for a couple of hours after seeing the answer to downvote it. –  O.L. Jun 27 at 16:21
    
What you say is totally untrue! If you beleave it, you should back it by data , or retract it. –  kjetil b halvorsen Jun 27 at 16:22

One function I would love to mention: The Greatest integer function or the Floor function.

$f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=[x]$ where $[x]$ denotes the greatest integer less than or equal to $x$

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oft-forgotten, yet oft-used, and usually poorly understood –  Moses Jun 28 at 7:30

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