Underbraces in mathematics - what do they mean?

[seriously disabled]

What do the underbraces in the following equations mean?

 

Edited May 30, 2010 by Uri

[Greippi]

I believe in this case it means something like "this also means/can be written as". Not entirely sure though.

[the tree]

In that context, that some term is equal to what is over the underbrace. Often they are used for general notes.

[seriously disabled]

Yeah that's what I was thinking intially but thanks for confirming it to me guys.

[Dave]

Usually they're informative. For example, people will often write something like

 

[math]\underbrace{1+1+\dots+1}_{\text{n times}}[/math]

 

which represents

 

[math]\sum_{k=1}^n 1[/math]

 

I find that it's a nice way of continuing steps in a proof without having to stop the train of thought, as it allows you to elaborate on a certain point in, say, a chain of equalities as is in your case.

[theoriginal169]

it seems like Plancherel s theorem you can exchange functions if f(x)=sigma(g(x)exp(ikx))

here you can exchange f and g while changing i to -i .