Kris
Thanks for responding
Let s = x+ iy
so -(s-s**2)**2 =
-( (x+iy) - (x+iy)^2 )^2 = -x^4 - 4 i x^3 y + 2 x^3 + 6 x^2 y^2 + 6 i x^2 y - x^2 + 4 i x y^3 - 6 x y^2 - 2 i x y - y^4 - 2 i y^3 + y^2
(symbolic calculator)
Suppose that -M < Re(s) < M, so above -y^4 is the leading real term for |y| large
so |exp(-(s-s**2)**2)| = O(exp(-y^4)) for -M < Re(s) < M
I need to do some contour integrals for Re(s) fixed, and over vertical lines of the form a + it.
I define g = lambda t: abs(G(2+j*t))
as t --> infty, the function does oscillate a little, but then it really decays fast
Here are the pointwise evaluations (below). When t = 3.8, g becomes
for l in range(1,100):
print(l*0.1,g(l*0.1))
3.3000000000000003 174581109.062865493774906
3.4000000000000004 310643.1174726955073363975
3.5 179.0204419216566700758697
3.6 0.03075815858756175907739449
3.7 0.000001446881510477273510161705
3.8000000000000003 1.707168282855990568833741e-11
3.9000000000000004 4.617468360109395918682982e-17
Now if I integrate the absolute value, it converges as it should
for l in range(1,50):
print('Integral from 0 to ', l*10, ' = ', quad(g,[0,l*10]))
Integral from 0 to 320 = 14283267314937294.78568836
Integral from 0 to 330 = 14283267221372905.29080994
Integral from 0 to 340 = 14283267263072427.82687573
Integral from 0 to 350 = 14283267349244040.12717721
Integral from 0 to 360 = 14283267186104994.47301923
Integral from 0 to 370 = 14283267319097570.887649
Integral from 0 to 380 = 14283267295148934.49766176
Integral from 0 to 390 = 14283267187229046.93666943
clearly it converges absolutely from the theory and the experimental evidence. But if I try to integrate the real or imaginary part, I get nonsense answers
redefine g = lambda t: re(G(2+j*t))
for l in range(1,50):
print('Integral from 0 to ', l*10, ' = ', quad(g,[0,l*10]))
Integral from 0 to 150 = -4027207776910521.878532395
Integral from 0 to 160 = 4475397256614113.025802875
Integral from 0 to 170 = 5032028392897261.000738395
Integral from 0 to 180 = 1368943760662856.81477378
Integral from 0 to 190 = -5670867742668200.379345995
Integral from 0 to 200 = 6293201245607762.71051154
Integral from 0 to 210 = -3030930189883354.986057916
Integral from 0 to 220 = -6843259753634191.086749538
Here is my code
#for l in range(1,1000):
# print(l*0.1,g(l*0.1))
for l in range(1,50):
print('Integral from 0 to ', l*10, ' = ', quad(g,[0,l*10]))