Indeed : ``` sage: f(x)=2^(x-1/2*I*log(-e^(-2*I*pi*x))/pi-1/2) ```
The key is probably ``` sage: f(x).diff(x) 0 ``` This should include an (infinite) series of terms in `dirac(x-k)` for k in integers... Logical consequence : ``` sage: f(x).integrate(x) 1/2*sqrt(2)*(-1)^(-1/2*I*log(2)/pi)*x ``` That is, your antiderivative is the product of `x` by aconstant, which turns out to be : ``` sage: f(x).integrate(x)/x 1/2*sqrt(2)*(-1)^(-1/2*I*log(2)/pi) sage: (f(x).integrate(x)/x).simplify() 1/2*sqrt(2)/(-1)^(1/2*I*log(2)/pi) sage: (f(x).integrate(x)/x).simplify_full() 1 ``` Note that `(2^floor(x)).integrate(x).plot((x,0,7/2))` gives : [image: tmp_0qt8ouke.png] which is correct. BTW : ``` sage: (2^floor(x)).integrate(x) 2^floor(x)*x ``` HTH, Le samedi 21 janvier 2023 à 15:34:11 UTC+1, Georgi Guninski a écrit : > I got an integral, which fails the derivative check. > > For real positive x, define > f(x)=2^(x - 1/2*I*log(-e^(-2*I*pi*x))/pi - 1/2) > f(x) is just an obfuscation of 2^floor(x) and > for all positive x, f(x) is integer. > Let g(x) be the indefinite integral of f(x) > and let gder(x)=g'(x). > Assuming correct computations, we should have > > gder(x)=g'(x)=f(x) (*) > > According to sage, gder is the constant $1$ > and (*) fails. > > sage session: > > === > f=2^(x - 1/2*I*log(-e^(-2*I*pi*x))/pi - 1/2) > g=integrate(f,x);gder=g.derivative(x) > g > #1/2*sqrt(2)*(-1)^(-1/2*I*log(2)/pi)*x > gder > #1/2*sqrt(2)*(-1)^(-1/2*I*log(2)/pi) > CC(gder) > #1.00000000000000 > x0=5;CC((f-gder)(x=x0)) > #31.0000000000000 > === > > Some questions: > 1. What other CASes say about g(x)? > 2. Why the derivative test fails? > 3. Besides the jumps at integer, do branches of log() > give instability? > > Some comments suggest discontinuous functions > cause integral problems. There are built-in > discontinuous functions like tan() which are > widely used. > > 4. Why tan() integrals are used without problems (?) > when this fails the derivative check? > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/0aa06411-e4e9-4485-a5e9-989cd2ea9327n%40googlegroups.com.
