I The product of 2 infinite sums

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dyn

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Hi.
I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
ΣnΣm xn/(n!) (-x)m/(m!)
without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ?
Thanks
 

fresh_42

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What do you get, if you do not ignore the ##i## as you did, and group the products by degree of ##x##?
 

scottdave

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Hi.
I know that eixe-ix = 1 but if I write the product of the 2 exponentials as infinite series I get
ΣnΣm xn/(n!) (-x)m/(m!)
without knowing the result is 1 using exponentials how would I get the result of this product of 2 infinite sums ?
Thanks
So the series for eu is Σn un/(n!)
and u is either ix or -ix (in your example). But you lost the i when converting to the series.
 

dyn

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Yes sorry I forgot the i when writing out the infinite series
 

WWGD

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You may also use: ##e^{ix}=(cosx+isinx)## and symmetry properties of ##sinx, cosx##.
 
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You may also use: ##e^{ix}=(cosx+isinx)## and symmetry properties of ##sinx, cosx##.
Sure, you could do that for this particular problem, but this ignores the OP's question about the product of two infinite series.
 

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