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Question: Is it possible to go from eq (2) to eq (7) using only Maple?

Question:Is it possible to go from eq (2) to eq (7) using only Maple?

mmcdara 7896 Maple
sum product
December 04 2023
2

Is it possible to transform relation (2) into relation (7) without using the hand-made relation (3) and the  Sum -> Int -> Sum trick?

> 

restart
> 

f := Product(x[i]^a*(1-x[i])^b, i)

Product(x[i]^a*(1-x[i])^b, i)

 (1)
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Lf := ln(f);

ln(Product(x[i]^a*(1-x[i])^b, i))

 (2)
> 

Sum(ln(x[i]^a*(1-x[i])^b), i)

Sum(ln(x[i]^a*(1-x[i])^b), i)

 (3)
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expand(%) assuming x[i] > 0, x[i] < 1, a > 0, b > 0

Sum(a*ln(x[i])+b*ln(1-x[i]), i)

 (4)
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eval(%, Sum=Int)

Int(a*ln(x[i])+b*ln(1-x[i]), i)

 (5)
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IntegrationTools:-Expand(%);

a*(Int(ln(x[i]), i))+b*(Int(ln(1-x[i]), i))

 (6)
> 

Lf = eval(%, Int=Sum)

ln(Product(x[i]^a*(1-x[i])^b, i)) = a*(Sum(ln(x[i]), i))+b*(Sum(ln(1-x[i]), i))

 (7)
> 

 

Download From2to7.mw

TIA
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