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Dec
29
comment Is there a symbol for integrating and setting $C=0$?
...Good example! And natural integral of it is neither of your two, it is $-\frac12 \cos(2x)$ wolframalpha.com/input/…
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel follow the link and insert 2x in the formula. wolframalpha.com/input/… And look at the result. U can insert any function and if the expression converges, u will receive the result in the "result" field
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel there are two formulas there, one uses Fourier transform, the other uses Newton series. You can also use pre-made tables or Wolfram Alpha: wolframalpha.com/input/… . (replace sin(x) with a function u want)
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel You can take any function with constant term and make from it a function without a constant term by simple operations. For instance, $\frac {x^3}3+2x+C=C(\frac{x^3}{3C}+\frac {2x}C+1)$
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel and what antiderivative you want? Any? Do u consider constant term is zero in $\ln ax$? Note that $\ln ax = \ln x + \ln a$. You can take any function with constant term and make from it a function without a constant term.
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel it is unclear what u want mathematically. If you want the antiderivative to be 0 in x=0, use integral from 0. The integral of 1/x is discontinuous in 0, so u choose a constant some other way. For instance, you can integrate from 1, then you will have $\ln x$. From the point of view of nature, though $\ln |x|+\gamma=\ln|x e^\gamma|$ is more natural (note that you can remove the constant by some elementary algebraic manipulations), so to get natural integral in a form without constant term, like $\ln|x e^\gamma|$.
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel For polynomials the first formula (integral from zero) will allways produce the same result as natural integral.
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel it will work for your condition that a function to be zero in zero. The function $1-\cos x$ is zero in zero, so it satisfies your requirement even though it seems more complicated than just $-\cos x$. If you want the most natural integration constant ever, there is no simpler way than to use the natural integral.
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@fvel the first variant, integral from zero to x will produce antiderivative that is zero in zero.
Dec
28
comment Is there a symbol for integrating and setting $C=0$?
@Henning Makholm yes, it is just a vestige of using the same formula for integrals/derivatives of other orders.
Dec
28
comment If my average speed is 50 mph, is it better to go 10% faster or 5 mph faster? (These are not the same)
It is better to go 10% faster.
Dec
26
comment What is the algebraic role of the mathematical constant $\gamma$?
@Alex R. well, but I thought I could miss something. My investigation so far was around the following: 1) $\gamma$ as natural integration constant when integrating $1/x$: natural integral of $1/x$ is $\gamma+\ln|x|$ 2) $\gamma$ as Cauchy mean value of some functions, such as $\Gamma(x)$ in poles, $\Gamma(0)=-\gamma$, which may have algebraic consequences following from functional equation for $\Gamma(x)$ 3) $\gamma$ as a twin of $\ln(4/\pi)$ (see wikipedia article)
Dec
26
comment What is the algebraic role of the mathematical constant $\gamma$?
So you say, it has no algebraic role? Of what ring $\gamma$ is identity element or idempotent or against what operation it is invariant, etc?
Dec
26
comment What is the algebraic role of the mathematical constant $\gamma$?
Okay. How is it related to algebra?
Dec
26
comment What is the algebraic role of the mathematical constant $\gamma$?
I see, so what? How this demonstrates the algebraic properties or $\gamma$? If I looked for analytic properties of expansions I could look ito Wikipedia. I am interested in alrgebraic role.
Dec
26
comment What is the algebraic role of the mathematical constant $\gamma$?
Can you show that these expansions demonstrate algebraic role of the constant? What the role is?
Dec
25
comment How to calculate $f(x)$ in $f(f(x)) = e^x$?
@Gottfried Helms typo. It does not converge when $a > e^{1/e}$, of course.
Dec
24
comment Closed form for sine graphic rotated by 45 degrees?
@Jack M this graph is wrong. It is rotated 60 degrees
Dec
24
comment Closed form for sine graphic rotated by 45 degrees?
@Henrik parametric expression is a closed form but for curve rather than for the function. Similarly, closed form for inverse function is not a closed form for straight function.
Dec
23
comment Closed form for sine graphic rotated by 45 degrees?
@Jack D'Aurizio yes.