Dieter Kaiser <drdieterkaiser at web.de> writes:
> Hello Rupert,
>
> do you have a more complete example what you would like to do?
>
> Dieter Kaiser
Hi,
Sorry for being vague. I'm attaching the code I was using at the bottom
of the email (yes, I know it's ugly).
The idea is that you're trying to evaluate a ratio of the form
integrate( topnop * p0 * s^(%i*x), s, 0, 1/ae )
/
integrate( bottom * s^(%i*x), s, 0, 1/ae )
You can do some asymptotic stuff (I hope) to realise that you can get
the answer in a power series of 1/x^n using just the values of the
integrands near 1/ae (since the oscillations cancel faster everywhere
else).
To evaluate this, you can't just do a Taylor series around 1/ae, since
Re*? and Ri*? aren't integers. So the idea was to expand both integrals
up to 1/ae - eps and then take the limit of the quotient as eps -> 0.
The zeroth order and the (disappearing) first order terms agree with
simulation. I haven't worked out how to get the next order - the limit's
too complicated for maxima to do without help I think. But anyway,
that's kind of irrelevant: what I would like to be able to do is say "p0
is a continuous function of s, guys, so please don't put noun forms in
your values for p00 etc."
Does that make sense? Sorry for not having a more minimal example to
hand.
Rupert
p0: ?*r0/( (1-ae*s)^(Re*?)*(1-ai*s)^(Ri*?) ) *
'integrate(
(1-ae*x)^(Re*?)*(1-ai*x)^(Ri*?)*(exp(x*vth)/(1-ae*x)-exp(x*vre)),
x, s, 1/ae )$
assume( Re*? > 0 )$
declare( [Re, ?, Ri, ?], noninteger )$
p00: limit(p0,s,1/ae)$
p01: limit(diff(p0,s),s,1/ae,minus)$
p02: limit(diff(p0,s,2),s,1/ae,minus)$
topnop: Re1*ae*(1-ae*s)^(Re*?-1)*(1-ai*s)^(Ri*?)$
bottom: (1-ae*s)^(Re*?)*(1-ai*s)^(Ri*?)*1/s*
( exp(s*vth)/(1-ae*s)-exp(s*vre) )$
nearb(expr) := at(expr, s=exp(-?)/ae)$
top0: nearb(topnop)*p00$
bot0: nearb(bottom)$
r10 = limit(top0/bot0, ?, 0, plus);
top1: nearb(diff(topnop,s))*p00 + p01*nearb(topnop)$
bot1: nearb(diff(bottom,s))$
ixr: 1/(%i*x)$
r11: limit( (ixr*top0 + ixr^2*top1)/(ixr*bot0 + ixr^2*bot1), ?, 0, plus );
'r11 = ratsimp(r11);
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