For varying values of n, I'd like to find values of a so that int_0^z
(x-a)^n dx has a root on the unit circle.
As I'd like high precision, mpmath seems right.
I can set things up as follows:
sage: from mpmath import *; mp.dps=30; mp.pretty=true
sage: var("a b c", domain="real")
sage: p=integral((x-a)^2, x)
sage: print p.subs(x=b+c*I).real()
sage: print p.subs(x=b+c*I).imag()
a^2*b - a*b^2 + a*c^2 + 1/3*b^3 - b*c^2
a^2*c - 2*a*b*c + b^2*c - 1/3*c^3
and then copy-and-paste the equations into a function
and solve:
sage: def f(a, b, c):
... return (a^2*b - a*b^2 + a*c^2 + 1/3*b^3 - b*c^2,
... a^2*c - 2*a*b*c + b^2*c - 1/3*c^3,
... b^2+c^2-1 )
...
sage: findroot(f, (0.57, 0.85, 0.49))
...
[0.577350269189625764509148780502]
[0.866025403784438646763723170753]
[ 0.5]
which works, but I need to eliminate the manual copy-and-paste.
What can I do to avoid the following error?
sage: def g(a, b, c):
... return (p.subs(x=b+c*I).real(),
... p.subs(x=b+c*I).imag(),
... b^2+c^2-1 )
...
sage: findroot(g, (0.57, 0.85, 0.49))
Traceback (most recent call last):
...
TypeError: g() takes exactly 3 arguments (1 given)
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