Use of LazyLaurentSeriesRing.

asked 2025-12-08 20:26:08 +0100

Peter Luschny
548 ●14 ●37 ●58

I would like to better understand the use of 'LazyLaurentSeriesRing' in Sage.

The expansion of the function 'Omega', as defined below, sometimes works, sometimes not.

More precisely, it works in the cases m = 0, 1, 2, 4, and does not work for m = 3, 5, 6.

What's the trick?

def Omega(m, z) :
    if m == 0: return z**0
    if m == 1: return exp(z)
    if m == 2: return cosh(z)
    if m == 3: return (exp(z) + 2 * exp(-z / 2) * cos(z * 3 ** (1 / 2) / 2)) / 3
    if m == 4: return (cosh(z) + cos(z)) / 2
    if m == 5: return (exp(z)
            + 2 * exp(-cos(pi / 5) * z) * cos(sin(pi / 5) * z)
            + 2 * exp(cos((2 * pi) / 5) * z) * cos(sin((2 * pi) / 5) * z) ) / 5
    if m == 6: return (cosh(z) + 2 * cos(z * sqrt(3) / 2) * cosh(z / 2)) / 3
    return 0

Test:

L = LazyLaurentSeriesRing(QQ, 'z')
for m in [0, 1, 2, 4]:
    f = Omega(m, L.gen())
    print(f"m={m}: {[factorial(n)*f.coefficient(n) for n in range(24)]}")

answered 2025-12-08 21:30:10 +0100

Max Alekseyev

7268 ●10 ●62 ●177

updated 2025-12-08 21:30:52 +0100

The problem is that you are mixing up symbolic values, like 3 ** (1 / 2), cos(pi / 5), etc., that do not exist in QQ and L.

A possible solution is to extend the base ring from QQ to AA to include radicals like sqrt(3), sqrt(5). However, explicit conversion of those expression into AA is needed:

def Omega(m, z) :
    if m == 0: return z**0
    if m == 1: return exp(z)
    if m == 2: return cosh(z)
    if m == 3: return (exp(z) + 2 * exp(-z / 2) * cos(z * sqrt(AA(3)) / 2)) / 3
    if m == 4: return (cosh(z) + cos(z)) / 2
    if m == 5: return (exp(z)
            + 2 * exp(-AA(cos(pi / 5)) * z) * cos(AA(sin(pi / 5)) * z)
            + 2 * exp(AA(cos((2 * pi) / 5)) * z) * cos(AA(sin((2 * pi) / 5)) * z) ) / 5
    if m == 6: return (cosh(z) + 2 * cos(z * sqrt(AA(3)) / 2) * cosh(z / 2)) / 3
    return 0

L = LazyLaurentSeriesRing(AA, 'z')
for m in (0..6):
    f = Omega(m, L.gen())
    print(f"m={m}: {[factorial(n)*f.coefficient(n).radical_expression() for n in range(24)]}")

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Comments

Excellent! Thank you so much!

Peter Luschny ( 2025-12-08 22:17:32 +0100 )edit

However, (1) the execution time is considerable, and (2) I would like to write a general function that doesn't require any special pre-processing of the input function. Perhaps 'LazyLaurentSeriesRing' simply isn't the right ring for this. Are there any better solutions?

Peter Luschny ( 2025-12-08 22:18:28 +0100 )edit

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