Substitution
By Olof Sisask (P3033) on Thursday,
April 26, 2001 - 05:44 pm :
Hi,
I was just wondering about integration by substitution: if you for example
say 'let
' - isn't that imposing a limit on
, i.e. it'll
always be between -1 and 1? Is it alright to do that if you substitute back
in the end?
Thanks,
Olof
By Anonymous on Friday, April 27,
2001 - 08:21 pm :
Hmmm... I suppose if you're integrating between -1 and 1 you haven't got a
problem anyway. Otherwise... you don't need a limit on
, do you, you just
need
to be a complex number and then
can take any value you
like... but can you integrate complex things?
By Olof Sisask (P3033) on Saturday,
April 28, 2001 - 11:24 pm :
I'm not sure how integration in the complex plane works?
By Anonymous on Sunday, April 29, 2001
- 06:47 pm :
Neither am I... that's why I left it as a question...
By Olof Sisask (P3033) on Sunday, April
29, 2001 - 10:10 pm :
Wasn't sure if you meant "can you" as in 'can I' or "can you"
as in 'can one'. Anyone have any ideas?
/Olof
By Anonymous on Sunday, April 29, 2001
- 10:27 pm :
Complex integration is an interesting
and extremely attractive part of mathematics.
Roughly speaking we generalise integration along the real line,
to integrating around paths in the complex plane. It is not to
difficult to explain and I am willing to go further if you are
interested.
Integration by subsitution - change of variable - rescales the
integral along the path of integration in the hope that the new
integral will be easier to evaluate.
By Olof Sisask (P3033) on Sunday, April
29, 2001 - 10:40 pm :
I'd be very interested in hearing about complex integration
Anon!
About integration by substitution - I understand the principle, but I'm not
sure about the justification. When you say that, for example,
,
then aren't you imposing a limit on
? I can see that if you adjust the
limits of the integral then this would be fine, but what's the justification
behind this when you're dealing with periodic functions like sin and cos? It's
probably one of those things where if I sat down and thought about it for a
while, it'd make sense.
Thanks,
Olof
By Anonymous on Sunday, April 29, 2001
- 11:33 pm :
You're right to be sceptical, proof of
the change of variable formula has to be done rigorously using
the axioms of the real numbers and such like.
You're correct to observe that x=cos(theta) restricts the range
of values of x - as you say we can only do a substitution if the
substituted expression can take all the values it needs to. Also
a change of variable function must be a one to one
correspondence.
By Olof Sisask (P3033) on Sunday, April
29, 2001 - 11:50 pm :
Ah that's what I was looking for, cheers Anon.