integrating a rapidly decaying function strange results

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J. Friedman

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Oct 30, 2018, 3:33:06 PM10/30/18
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According to my calculations g(t) (below) should be very rapidly decaying as t \rightarrow \infty, but the integral (quad) seems to give me strange results. Any suggestions on what is going on?

G = lambda s: exp(-(s-s**2)**2)
g = lambda t: re(G(2+j*t))
for l in range(1,10):
    print(quad(g,[0,l*2]))

-929368047261.93620154083404308518186452893823837753
0.73030807681355697617926688010183740605274248154493
0.73030807681355697617926682956789024942627475882531
0.73084346535653543017375370300762616576048495881376
6506.1108968477591798125903527040968078450818858557
-10497334.097217461344387940173023804897764062383082
199049954.0344803451280316219162110047797808274189
10165944034.061867509342219271383876244817897781579
-90252651558.490101003579618384469577487140317126004




Kris Kuhlman

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Oct 31, 2018, 4:23:50 AM10/31/18
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That function is only rapidly decaying for real s. It is oscillatory for complex arguments, as you show here.

Try plotting the function.

Kris

On Tuesday, October 30, 2018 at 1:33:06 PM UTC-6, J. Friedman wrote:
> According to my calculations g(t) (below) should be very rapidly decaying as t \rightarrow \infty, but the integral (quad) seems to give me strange results. Any suggestions on what is going on?
>
>
>
> G = lambda s: exp(-(s-s**2)**2)g = lambda t: re(G(2+j*t))

J. Friedman

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Oct 31, 2018, 9:43:49 PM10/31/18
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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.


G = lambda s: exp(-(s-s**2)**2)

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


G = lambda s: exp(-(s-s**2)**2)
g = lambda t: re(G(2+j*t))

#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]))
   




   








Kris Kuhlman

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Oct 31, 2018, 9:53:43 PM10/31/18
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Real and imaginary parts are both oscillatory, when real part is positive.

I wish mpmath had an iPhone app.
B0CCCC5B-48CD-42AB-95E3-791109B3FDDE.png
8C2C0137-0417-456C-81FA-E05110473DAF.png
27B72790-3710-4AD4-8356-061AB141D3ED.png

Joshua Friedman

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Oct 31, 2018, 10:16:25 PM10/31/18
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Yes, I see that it oscillates, but the integral converges absolutely. From the plots you sent, when |y| > 5, it is rapidly decreasing (in absolute value).  Does the mp math quad integral not work on oscillating functions, even if they decay in absolute value?

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Kris Kuhlman

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Oct 31, 2018, 10:20:09 PM10/31/18
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http://www.mpmath.org/doc/current/calculus/integration.html

The quadosc function works with oscillating functions (like J Bessel functions)

You either need to supply the person or the zeros for it to work.
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