0
votes
3answers
51 views

How to get power by knowing the number and result

How to get power by knowing the number and result. For Example $$2^n = 8$$ how can i return the power $n$ by knowing number $2$ and result $8$ or $$4^n = 1024$$ how can i return the ...
1
vote
3answers
154 views

Expand into power series $f(x)=\log(x+\sqrt{1+x^2})$

As in the topic, I am also supposed to find the radius of convergence. My solution: $$\log(x+\sqrt{1+x^2})=\log \left ( x(1+\sqrt{\frac{1}{x^2}+1})\right )=\log(x)+\log(1+\sqrt{\frac{1}{x^2}+1})$$Now ...
1
vote
0answers
77 views

Use of natural logarithm transformation on weighted index series

I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$. The exponent represents the weight for variables, $x, y$ and $z$ in the example above. Applying natural logarithm on $x^5+y^8+z^2$, I get ...
1
vote
0answers
76 views

What are the Puiseux series for the inverse of $exp(z)(z-a)(z-b)(z-c)$ expanded at the singularities?

Let $a,b,c$ be real variables. Let $z$ be a complex number and $g(z) = exp(z)(z-a)(z-b)(z-c)$. Let $f(z)$ be the functional inverse of $g(z)$ such that $f(g(z)) = g(f(z)) = z$. Now $f(z)$ must have ...
0
votes
1answer
386 views

Try to find an approximation by logarithm function.

Recently I am thinking about this question: Assume $x$ is real, $x\geq0$, $c$ is a positive constant number and $z$ is also a real constant between $3.5$ and $4$. Now there is a function: $$ ...
1
vote
3answers
110 views

Formula to $\ln$ that holds on interval $x \geq 1$

In the Wikipedia we can find two formulas using power series to $\ln(x)$, but I would like a formula that holds on the interval $x \geq 1$ (or is possible to calculate $\ln(x)$ to $x \geq 1$ with the ...
5
votes
3answers
236 views

Summing up the series $a_{3k}$ where $\log(1-x+x^2) = \sum a_k x^k$

If $\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots = $ ? My approach is to write $1-x+x^2 = \frac{1+x^3}{1+x}$ then expanding the respective logarithms,I got a series ...