Linked Questions
10 questions linked to/from Series solution of the trinomial equation
8
votes
3
answers
1k
views
Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?
This seems such a simple question that I fear I must have missed some elementary maths.
I am looking for a way to solve $x+x^a = y$ by reference to an already defined function, $a,x,y > 0$ are real....
- 83
8
votes
3
answers
1k
views
Series solution for general trinomial
Consider the equation
$x^5-2x^2+z=0$
How do you derive the Lagrange inversion theorem series solution for it? I know it exists because the answer is here for any trinomial https://arxiv.org/pdf/0910....
- 367
2
votes
2
answers
1k
views
Transcendental formulas for roots of polynomials
It is well known that there is no general formula for the roots of a polynomial of degree $\geq$ 5 solely in terms of arithmetical operations and radicals.
Are there formulas using other kinds of ...
- 29
6
votes
2
answers
403
views
What's the summation of formal series $\sum_{n\geq0}\binom{n\delta}{n}x^n$？
$\delta$ is a positive number. Is this Taylor expansion of some function?
- 61
1
vote
2
answers
1k
views
Bounds on the smallest real positive root of a polynomial
I'm trying to find upper and lower bounds of the smallest positive root of a polynomial, stated in terms of its coefficients. As I appreciate it might be a very general problem, My specific interest ...
- 3,574
2
votes
3
answers
259
views
How to show continuity and monotonicity of solutions to this parametrized equation?
Let $1 \le p <2$ be a parameter. Consider the equation
$$
\frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1}
$$
I am rather certain that for each $1 \le p <2$, ...
- 6,741
3
votes
2
answers
216
views
Showing that the inverse of a function is approximately equivalent to $\frac{1}{n^{1/\alpha}}$
I'm currently working with someone on my PhD, and last week they asked me to check that a certain approximation holds as an exercise. Unfortunately, I couldn't figure out how to do it, and we've since ...
0
votes
2
answers
224
views
Solutions or bounds for $x^{(1-\epsilon)/2}=1-x$
Is there a known expression for the solution of the following simple equation, or at least good bounds on the solution? Assume $\epsilon \in [0,1)$ is a given parameter and $x \in (0,1)$.
$$x^{(1-\...
- 1,324
0
votes
1
answer
298
views
Implicit inequality
Let $A,B,d\ge 1$ and suppose that $x\ge0$ satisfies
$$ x^{\frac{d+1}{d}}
\le
Ax+B.
\qquad(*)
$$
I can show that $(*)$ implies the bound
$$
x< d(A^d+B).
\qquad(**)
$$
Questions: (1) Can a ...
- 6,541
3
votes
0
answers
289
views
Functional inverse of $z=1+w+\cdots+w^{n-1}$
Migrated from the MSE.
I am interested in the functional inverse of
$$
z=1+w+\cdots+w^{n-1},\quad w\geq0,\ n>1.
$$
This function is strictly increasing on $w\geq0$ and thus admits an inverse.
By ...
