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 Dec29 asked Why formal power series are not considered a system of hypercomplex numbers? Dec29 answered Is there a symbol for integrating and setting $C=0$? Dec29 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/… Dec28 revised Is there a symbol for integrating and setting $C=0$? added 249 characters in body Dec28 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 Dec28 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) Dec28 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, $\frac {x^3}3+2x+C=C(\frac{x^3}{3C}+\frac {2x}C+1)$ Dec28 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. Dec28 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 $\ln|x e^\gamma|$. Dec28 revised Is there a symbol for integrating and setting $C=0$? deleted 5 characters in body Dec28 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. Dec28 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. Dec28 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. Dec28 revised Is there a symbol for integrating and setting $C=0$? deleted 12 characters in body Dec28 revised Is there a symbol for integrating and setting $C=0$? deleted 6 characters in body Dec28 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. Dec28 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. Dec28 answered Is there a symbol for integrating and setting $C=0$? Dec26 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) Dec26 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?