In mathematics, Euler's identity (also known as Euler's equation) is the equality
where
e is Euler's number, the base of natural logarithms,
i is the imaginary unit, which satisfies i2 = −1, and
π is pi, the ratio of the circumference of a circle to its diameter.
Euler's identity is named after the Swiss mathematician Leonhard Euler. It is considered an example of mathematical beauty.
== Explanation ==
Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x,
where the values of the trigonometric functions sine and cosine are given...
user61230
In mathematics, the look-and-say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ... (sequence A005150 in OEIS).
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11.
11 is read off as "two 1s" or 21.
21 is read off as "one 2, then one 1" or 1211.
1211 is read off as "one 1, then one 2, then two 1s" or 111221.
111221 is read off as "three 1s, then two 2s, then one 1" or 312211...
Prove by definition that: $$ \lim_{(x,y)\to(1,1)} \frac
{x^2+2xy-3y^2}{x^2-y^2} = 2 $$
I've been reading quite a lot about how prove this limit, so I want to show what I've done so far so you can tell me any suggestions (tricks) and even point out any mistake.
What I've tried:
I want to f...
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Q: About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $
Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them. At some point I'll add my solution.
It's a question for the informative purpose rather than finding solutions, the solution ...
