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52 lines
1.7 KiB
52 lines
1.7 KiB
def finite_diff(expression, variable, increment=1):
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"""
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Takes as input a polynomial expression and the variable used to construct
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it and returns the difference between function's value when the input is
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incremented to 1 and the original function value. If you want an increment
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other than one supply it as a third argument.
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Examples
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========
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>>> from sympy.abc import x, y, z
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>>> from sympy.series.kauers import finite_diff
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>>> finite_diff(x**2, x)
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2*x + 1
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>>> finite_diff(y**3 + 2*y**2 + 3*y + 4, y)
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3*y**2 + 7*y + 6
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>>> finite_diff(x**2 + 3*x + 8, x, 2)
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4*x + 10
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>>> finite_diff(z**3 + 8*z, z, 3)
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9*z**2 + 27*z + 51
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"""
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expression = expression.expand()
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expression2 = expression.subs(variable, variable + increment)
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expression2 = expression2.expand()
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return expression2 - expression
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def finite_diff_kauers(sum):
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"""
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Takes as input a Sum instance and returns the difference between the sum
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with the upper index incremented by 1 and the original sum. For example,
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if S(n) is a sum, then finite_diff_kauers will return S(n + 1) - S(n).
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Examples
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========
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>>> from sympy.series.kauers import finite_diff_kauers
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>>> from sympy import Sum
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>>> from sympy.abc import x, y, m, n, k
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>>> finite_diff_kauers(Sum(k, (k, 1, n)))
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n + 1
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>>> finite_diff_kauers(Sum(1/k, (k, 1, n)))
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1/(n + 1)
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>>> finite_diff_kauers(Sum((x*y**2), (x, 1, n), (y, 1, m)))
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(m + 1)**2*(n + 1)
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>>> finite_diff_kauers(Sum((x*y), (x, 1, m), (y, 1, n)))
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(m + 1)*(n + 1)
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"""
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function = sum.function
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for l in sum.limits:
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function = function.subs(l[0], l[- 1] + 1)
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return function
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