# Tagged Questions

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 $(+, –, ×, ÷)$.

57 views

### Is the following statement provable?

The statement is: $f$ is a real fucntion on $\mathbb R$. Then if $f'(x)=f(x)$ and $f(0)=1$, then $f(x)\neq 0.$
31 views

### Integral of a product of a cosine function with argument x and a confluent hypergeometric function with argument $x^2$

I'm trying to integrate the following $$\int_0^1{_1 F_1 (a;2a;i \alpha x^2)}\cos{\beta x}dx$$ where $\alpha$ and $\beta$ are real numbers.
108 views

### Polynomials with degree $5$ solvable in elementary functions?

Quadratic, cubic and quartic polynomials are solvable in radicals, so there is no question here. What about the polynomials of degree $5$ (quintic)? Do we know all such polynomials (classes of ...
40 views

### Graphical explanation of the difference between $C^1$ and $C^2$ function?

We are all aware of the intuitive (graphical) explanation of the concepts of continuous and differentiable function. Whenever these two concepts are formally defined, the following elementary ...
44 views

30 views

### Prove $f: A \rightarrow B$ is strictly injective, $\implies$ $f^{-1}$ is a function and dom $f^{-1} \subset B$

The question I have about this proof is that, do I need to choose a specific function $f:A\rightarrow B$ that is not injective but surjective? Will I lose generality if I do? For instance, I was ...
34 views

20 views

### How to show that: $\log_a (x^{a}-x)-\log_a \Big(\dfrac{x^{a}-x}{a}\Big)=1$, where $a$ and $x$ are positive integers.

I was studying Fermat's Little Theorem and Logarithm to see if there is any interesting result or correlation exist between the two. So I came up with this equation. I know few basic logarithmic ...
47 views

38 views

### Elementary Functions, Differentiation, Integration [duplicate]

Why is it that differentiation of a function that is a composition of elementary functions (such as $\sin \:2^x$ or $\ln(\mathrm{arcsec}\: x^3)$ or $x^{1/x}$) always produces a composition of ...
47 views

### Any smoother version of the exponential function?

Often one needs to express some quantity of interest in a scale other than its original one. One can use the exponential function to map $(-\infty,0)\to(0,1)$ and $(0,+\infty)\to(1,+\infty)$, but ...
61 views

33 views

### Power series for non elementary functions

Since the function $f(x)=e^{-x^2}$ cannot be integrated using elementary functions, how could one find a power series for $F$, where $F$ is an elementary function such that $F'(x)=e^{-x^2}$?
41 views

### Is $f([a]_{mn}) = ([a]_m,[a]_n)$ a bijection?

Given, $f : Z/mnZ → Z/mZ × Z/nZ$, is $f([a]_{mn}) = ([a]_m,[a]_n)$ a bijection? I have already done the work to prove that this function is well-defined. Can I say that this is bijective though, ...
42 views