For questions on elementary functions, functions of one variable built from a finite number of exponentials, logarithms, constants, and $n$th roots through composition and combinations using the four elementary operations $(+, –, ×, ÷)$.

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3
votes
0answers
21 views

Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
0
votes
2answers
37 views

Asymptotic approximation of the arctangent?

That is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x)) = ...
1
vote
0answers
34 views

Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?

Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?
0
votes
0answers
35 views

Monotonically decreasing function for multiplication product?

I have a set of numbers $S = [100,999]$ for which I want the maximum product $p$ such that $p = a \times b$ for all $a,b \in S$ also fulfilling some condition $C$. I would like $p$ to be monotonically ...
2
votes
1answer
483 views

Need to prove that the equation has only 1 solution.

I have been trying to solve the following equation: $5^x+7^x=12^x.$ Obviously, x=1 is a solution but how do I prove that there are no other solutions.
3
votes
2answers
74 views

prove a sequence without knowing its convergence

Let $(X_n)$ be the sequence with $X_1=2$ and $X_n=\sqrt{5X_{n-1} + 6}$ for all $n\ge 2$. How can you prove that it is convergent? IF given convergence, I know that its limit is $6$, but the question ...
1
vote
2answers
21 views

Inequality Expression

what happens when an inequality expression is divided by the same number. Did the sign stay the same each time or change directions? If the sign changed, explain what made the direction of the sign ...
0
votes
0answers
33 views

Autocorrelation of Raised Cosine Function

Let us define the raised cosine function as follows: $f \left( x \right) = \dfrac{\left( 1 + \cos \left( x \right) \right)}{2}$, for $- \pi < x < \pi$. $f \left( x \right) = 0$, elsewhere. I ...
0
votes
2answers
35 views

Evaluate $\lim_{x\to 0}\frac{1}{x^4}\int_{-x}^{0}\sin(t^3)\,dt$

Evaluate $$\lim_{x\to 0}\frac{1}{x^4}\int_{-x}^{0}\sin(t^3)\,dt.$$ I use L'Hopital's Rule and get -1/4. The solution says 1/4. Any ideas?
5
votes
1answer
116 views

Is there a theory of integration in elementary terms for definite integrals?

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For ...
3
votes
2answers
41 views

Solution of $(1-x)^p= x$ in $(0,1/n)$

Let $n,p$ be positive integers. The equation $$ (1-x)^p = x $$ has a unique solution $x_p$ in the interval $(0,1)$. This follows by the monotonicity properties of $(1-x)^p$ and $x$. My question is: ...
2
votes
1answer
63 views

Injective map from real projective plane to $\Bbb{R}^4$

Consider the mapping $\Bbb R^3\rightarrow\Bbb R^4$ given by $$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$ which passes to the quotient and can therefore be viewed as a map from the projective plane $\Bbb ...
9
votes
11answers
416 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
3
votes
4answers
74 views

Find the derivative of $y=x\sqrt{9-x}$

"Find the derivative of $y=x\sqrt{9-x}$." So this is what I have and now I'm stuck. \begin{align} y' &= x \frac{d}{dx}\left[(9-x)^{1/2}\right] + (9-x)^{1/2} \frac{d}{dx}(x)\\ &= x ...
0
votes
0answers
63 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
1
vote
0answers
39 views

Power series solution for a DE with Frobenius method

The given DE is $(x²-3)y"+2xy'=0$ Since there is a singular point ($x=\pm\sqrt{3}$) I used the Frobenius method. I found two indicial relationships: $-3r(r+1)=0$ and $-3(r+1)(r+2)=0$ because I have ...
1
vote
1answer
17 views

Composition of mappings on finite sets

If I'm working in the realm of finite sets on the form $\underline{n} = \{1, \ldots, n\}, n \in \mathbb{N} $. Consider any two transformations $f :\underline{n} \to \underline{m}$ and $g ...
0
votes
1answer
37 views

Find a polynomial with certain conditions.

Suppose that: $$f(x) = 3\frac{x^4+x^3+x^2+1}{x^2+x-2}.$$ Find a polynomial $h(x)$ such that $f(x) + h(x)$ has horizontal asymptote of 0 as $x$ approaches positive infinity.
1
vote
0answers
35 views

Is it possible to find set of functions so that their antiderivatives are also in this set?

Antiderivatives of elementary functions sometimes are not elementary functions. Is it possible to add finite number of non-elementary functions to elementary function set so that their ...
1
vote
4answers
54 views

Equation , powers of two

I want to find the sum of the roots of the equation $$4(4^x + 4^{-x}) - 23(2^x + 2^{-x}) + 40 = 0 $$ in real numbers. I tried the substitution $ 2^x = t $ but then it turns into a quartic equation ...
26
votes
0answers
491 views

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
37
votes
4answers
2k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
1
vote
0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
3
votes
3answers
125 views

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$
1
vote
0answers
61 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
1
vote
0answers
45 views

Repository of functions which do not have elementary integrals [duplicate]

If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression. One ...
4
votes
2answers
76 views

value of $a+b$ of the following function

$f(x)=x^3-3x^2+5x\;$ and $\;f(a)=1,f(b)=5.\;$ Find $a+b$. I know only one real root exist for each equation as derivative of the function is always positive .I do not intend to use the formula of ...
1
vote
0answers
13 views

efficiently solve for values of a coefficient in a function, so for those values, the function intersects another function a specific number of times.

This is my summer assignment for my freshman "Intro to Numerical Methods with Matlab: Unit 2" course. The task: "Write an efficient Matlab code, which will take any closed $f(x)$ and $g(x)$ and ...
2
votes
2answers
80 views

Properties of $f(a)-f(b)=a-b$

Let: $f(a)-f(b)=a-b$ and $f(a)>f(b)$ and $f(a),f(b)>0$ and $a>b$ Based on the above facts is it sufficient enough to say that $f(a)=a$ and $f(b)=b$?
2
votes
3answers
69 views

Solving equation involving logarithm

How do I find $x$ in the equation $x - \ln x = 1.9$? Next I have $x - 1.9 = \ln x$ we are learning about logarithm but as I tried to take $\ln$ of both side it leads me to nothing. Could you help ...
0
votes
1answer
42 views

How to define a continuous function $f:I\times I\longrightarrow I$ such that $f(0, t)=t$ and $f(1, t)=1$ for all $t\in I$?

Let $I=[0, 1]$. I need some help to define a continuous function $f:I\times I\longrightarrow I$ such that $$f(0, t)=t\quad \textrm{and}\quad f(1, t)=1$$ for all $t\in I$. The nearest I got was: $$f(s, ...
1
vote
1answer
132 views

What does “versus” mean in the context of a graph?

Say you have a graph of say $y=mx+b$, with $x$ on the horizontal axis and $y$ on the vertical axis. You need to give the graph a title, would you say: This is a graph of "$y$ versus $x$?" or This ...
1
vote
1answer
39 views

Inverse of Higher logarithms

Th polylogarithm function is defined by $$Li_s(z)=\sum_{k=1}^\infty\frac{z^k}{k^s}.$$ At $s=1$, we have the natural logarithm function. We have the inverse of natural logarithm function as the ...
0
votes
1answer
23 views

How to work out the probability of 2 people having a different birthday

I'm trying to reproduce the 'birthday problem' where you work out the probabilities of n people having the same / different birthday. Theres a good example here: ...
0
votes
1answer
61 views

How to graph a sin & cos wave

I've got a time column of numbers(x) from 0 - 20 in 0.1 intervals and i've got a sin(x) and cos(x) column. I've got to produce some kind of wave like graph from it. Although I'm not sure which numbers ...
0
votes
1answer
56 views

Inverse of $a f(x)$ and inverse of $a f(x) + b$

Is there a general rule for the inverse of the function $ g(x) = a f(x) $, where $a$ is a constant, assuming $f^{-1}(x)$ is defined? Follow up: $g(x) = a f(x) + b$. Is the following correct, given ...
0
votes
2answers
26 views

Distributive property

I have the following question on distributive property. If I multiply $3$ to the given expression $\frac{1}{3}\pi r^2h$. Question: $3\cdot\frac{1}{3}\pi r^2h$ Based on what I understand for ...
8
votes
0answers
57 views

Is there a name for the class of functions which are infinitely integrable in elementary functions?

Is there a name for the class of functions which are infinitely integrable in elementary functions, that is whose consecutive integrals also elementary not depending on how much times we took the ...
0
votes
0answers
43 views

Why this function is elementary and its pair is not?

Why $$f(x)=\frac{2 \zeta (2)}{\pi ^2}+\frac{6 \zeta(4) x^2}{\pi^4}+\frac{10 \zeta(6) x^4}{\pi^6}+\frac{14 \zeta(8) x^6}{\pi^8}+\cdots$$ is elementary while $$g(x)=\frac{4 \zeta (3)x}{\pi ...
2
votes
2answers
94 views

Liouville's Criterion and Integration of $\sin(z)/z$ using elementary functions

Liouville's Criterion: $\int fe^g$ is elementary iff there is rational function $q$ in $\mathbb{C}(z)$ such that $$f=q'+qg'$$ where $f\neq0$ and $g$ is a non constant function in $\mathbb{C}(z)$. Now ...
0
votes
1answer
23 views

$T: [0,1)\to [0,1), x\mapsto 10x-\lfloor 10x\rfloor$

Consider $T: [0,1)\to [0,1), x\mapsto 10x-\lfloor 10x\rfloor$. Is that the same as $$ 10x (mod 1)? $$ or in which sense is that multiplication with 10 mod 1?
8
votes
0answers
178 views

Is each “elementary + finite functions” function “elementary + finite functions”-integrable?

It is known that there exists elementary functions which are not elementary integrable, i. e. there exists no elementary anti derivative. Example: $f(x) = e^{-x^2}$. Let $A$ be the set of elementary ...
4
votes
2answers
374 views

List of functions not integrable in elementary terms

When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...
2
votes
1answer
18 views

Simultaneous Equation (I think)

I am not sure whether I am just not remembering the technique or I don't have enough clues to solve this one: $T_1 - T_2 = 362$ $\frac{T_1}{T_2} = 5.48$ I cannot seem to solve for $T_1$ or $T_2$ I ...
0
votes
0answers
57 views

max min sum product

I am confused with this elementary thing: For all real $x,y$ define $\vee, \wedge$ by $x\vee y=\max\{x,y\}$ and $x\wedge y=\min\{x,y\}$. I am going to know the closed formula of sum and product ...
7
votes
2answers
129 views

What is the mathematical relevance of whether an expression has a closed form?

In the evaluation of mathematical expressions, particularly integrals, I often find a statement that the expression has or does not have a closed form. I looked up the definition, and the important ...
0
votes
1answer
24 views

Would this function be correct?

I want to express a function that for any x input, it outputs the nearest EVEN integer less than or equal to x. Would $g(x) = \{ \lfloor x \rfloor : 2 \mid x \} $ do the job properly? Read: $g(x)$ ...
3
votes
1answer
178 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int ...
0
votes
0answers
28 views

About a specific mathematical series which is a power of the exponential function

My professor wrote the below exponential function just out of the box when he suggested a kernal for a 1D domain. $f(x) = e^{-\Big(a_1x_1+ \dfrac{1}{2} a_2 x_2^2 + \dfrac{1}{3} a_3 x_3^3 + .... ...
0
votes
3answers
111 views

Equation including tangent function

I've been studying one problem and I need to consider the following problem. Let $n\in\mathbb{N}$. Is $\tan\left(nx\right)=n \tan(x)$ solvable on $\left(0,\pi\right)$? If it is, what are solutions?