Integral of exp(sin(x)) dx
Chris Rodgers
I have a problem which involves finding integrals of the form
/ x=X
|
| exp(sin(x)) dx
|
/ x=0
or
/ x=X
|
| exp(a+b*x+sin(c*x)) dx
|
/ x=0
where a,b,c are real numbers.
Does anyone have any suggestions on how such an integral might be
tackled symbollically? I have already tried Maple and Mathematica on
these with no success.
Alternatively, is there a compelling reason why such integrals CANNOT be
found (as opposed to my just not being smart enough to do them)?
Many thanks,
Chris Rodgers
http://rodgers.org.uk/
Dirk Van de moortel
"Chris Rodgers" <rod...@physchem.NOSPAMox.aREMOVEc.uk> wrote in message news:eo35b5$rhv$1...@frank-exchange-of-views.oucs.ox.ac.uk...
Because some such integrals are insufficiently interesting.
It's like int{ sin(x)/x dx }.
This cannot be found either, but this one is interesting,
so we defined it as Si(x) and call it the "Sine integral of x"
http://mathworld.wolfram.com/SineIntegral.html
It could also be like int{ 1/x dx }.
This cannot be found either, but being extremely interesting,
we can define it as ln(x) and call it "the natural logarithm of x".
There are of course other ways to define ln(x).
What do you need int{ exp(sin(x)) dx } for?
Dirk Vdm
Dave Seaman
> or
You might consider expanding the function in a Taylor series and
integrating termwise.
Mathematica 5.2 for Mac OS X
Copyright 1988-2005 Wolfram Research, Inc.
-- Terminal graphics initialized --
In[1]:= ess = Series[Exp[Sin[x]],{x,0,9}]
2 4 5 6 7 8 9
x x x x x 31 x x 10
Out[1]= 1 + x + -- - -- - -- - --- + -- + ----- + ---- + O[x]
2 8 15 240 90 5760 5670
In[2]:= Integrate[ess,x]
2 3 5 6 7 8 9 10
x x x x x x 31 x x 11
Out[2]= x + -- + -- - -- - -- - ---- + --- + ----- + ----- + O[x]
2 6 40 90 1680 720 51840 56700
--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Chris Rodgers
It's for some quantum mechanics. The sin(x) arises because I apply an
oscillating field which introduces sin(const * t) terms into the system
energy levels. The exponential is for the quantum mechanical time
evolution of a state in this system. It is very easy to calculate all
this numerically, but not very informative on "why" things happen the
way they do. I was hoping that a symbolic result might shed more light
on things.
Chris.
Rob Johnson
In article <r99ph.283550$N77.5...@phobos.telenet-ops.be>,
Since natural logarithms had been around about 100 years or so before
calculus, the integral of 1/x was probably never considered unsolvable
in elementary terms. I understand your point, but log(x) is not like
Si(x) in the sense that Si(x) is defined as an integral, while log(x)
was originally defined as the inverse of the exponential function.
>What do you need int{ exp(sin(x)) dx } for?
If necessity were the only parent of mathematics, we wouldn't have
nearly the technology we have today. Pure curiosity has proven a
good reason for many questions. Applications will follow as needed.
In any case, the question of which functions have an anti-derivative
seems pretty interesting to me.
Rob Johnson <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
Herman Rubin
>I have a problem which involves finding integrals of the form
> / x=X
> |
> | exp(sin(x)) dx
> |
>/ x=0
>or
> / x=X
> |
> | exp(a+b*x+sin(c*x)) dx
> |
>/ x=0
>where a,b,c are real numbers.
>Does anyone have any suggestions on how such an integral might be
>tackled symbollically? I have already tried Maple and Mathematica on
>these with no success.
For good reason; it cannot be done.
>Alternatively, is there a compelling reason why such integrals CANNOT be
>found (as opposed to my just not being smart enough to do them)?
There is a means of deciding whether an anti-derivative
can be computed in closed form. It is not always easy
to carry out.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Robert Israel
Herman Rubin <hru...@odds.stat.purdue.edu> wrote:
>In article <eo35b5$rhv$1...@frank-exchange-of-views.oucs.ox.ac.uk>,
>Chris Rodgers <rod...@physchem.NOSPAMox.aREMOVEc.uk> wrote:
>>Hi,
>
>>I have a problem which involves finding integrals of the form
>
>> / x=X
>> |
>> | exp(sin(x)) dx
>> |
>>/ x=0
>
>>or
>
>> / x=X
>> |
>> | exp(a+b*x+sin(c*x)) dx
>> |
>>/ x=0
>
>>where a,b,c are real numbers.
>
>>Does anyone have any suggestions on how such an integral might be
>>tackled symbollically? I have already tried Maple and Mathematica on
>>these with no success.
>
>For good reason; it cannot be done.
>
>>Alternatively, is there a compelling reason why such integrals CANNOT be
>>found (as opposed to my just not being smart enough to do them)?
>
>There is a means of deciding whether an anti-derivative
>can be computed in closed form. It is not always easy
>to carry out.
antiderivative that is an elementary function. That
is a precise term, while "closed form" is rather nebulous,
depending on which special functions you take as "known".
You might look at
<http://www.math.niu.edu/~rusin/known-math/97/nonelem_integr2>
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Dirk Van de moortel
"Rob Johnson" <r...@trash.whim.org> wrote in message news:2007011...@whim.org...
Indeed, that is why with the ln-example I explicitly used phrases like
"It *could* also be like..."
and
"we *can* define"
:-)
>
>>What do you need int{ exp(sin(x)) dx } for?
>
> If necessity were the only parent of mathematics, we wouldn't have
> nearly the technology we have today. Pure curiosity has proven a
> good reason for many questions.
Sure, I was just wondering where the thing occured. It wasn't
meant to be derogatery in any way. On the contrary.
> Applications will follow as needed.
> In any case, the question of which functions have an anti-derivative
> seems pretty interesting to me.
Yes.
Dirk Vdm
G. A. Edgar
Rodgers <rod...@physchem.NOSPAMox.aREMOVEc.uk> wrote:
> / x=X
> |
> | exp(sin(x)) dx
> |
> / x=0
>
Certain definite integrals (X = 2 pi maybe) can be done
in terms of Bessel functions. The indefinite integral is
not elementary: substitute u=sin(x) to get
integral(exp(u) du /sqrt(1-u^2)) , and this should
fit the Liouville framework to determine this.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Dirk Van de moortel
"Chris Rodgers" <rod...@physchem.NOSPAMox.aREMOVEc.uk> wrote in message news:eo3aaa$tfp$1...@frank-exchange-of-views.oucs.ox.ac.uk...
Okay. Good thinking.
Alas, I'm afraid there's no closed form in this case.
That is, unless of course you define a new closed form function
and convince the major calculator manufactures to put it between
the natural log and the sine ;-)
Cheers,
Dirk Vdm
user923005
Since it appears that this simple looking integral is very difficult to
solve symbolically, it might be worthwhile to try Borwein's technique
of numerical integration to 1000 digits or so and then using PSLQ to
look for patterns.
http://www.aarms.math.ca/events/atlantic/Presentations/jborwein.pdf
It sounds like a very interesting problem. Let us know if you find a
solution.
Jon Slaughter
Sure. There are many functions that do not have an "elementary"
anti-derivative. See http://en.wikipedia.org/wiki/Differential_Galois_theory
.
Usually in this case one "artificially" defines the anti-derivative to be
the anti-derivative of that function. i.e., its really for symbolic reasons
and to express over functions in terms of the anti-derivative in a
simplified way(so you don't have to write the integral out every time).
In a sense though even functions like sin(x) are not elementary and have no
elementary derivatives. For example, compute int(sin(x),x=0..1) exactly?
well its obviously cos(1) but what is cos(1)? Do you know exactly what it
is? (maybe, there are many formulas to find many "special" values of the
trig functions but AFAIK there are an infinite number of arguments that do
not produce exact results from finite means(you can compute them with
infinite series but in that sense they are not elementary). Nevertheless we
still think of integrals of these trig functions as elementary because they
are so common and do have special properties that makes them easy to work
with.
(I'm not sure if the above is entirely correct but I think it gets the jist
of it. I think it all boils down to being able to compute the values at had
using a finite number of additions(when all operations are converted to
addition). If you can't then its not elementary. (I could be wrong here but
this is how I tend tot hink of it))
Jon
Dave L. Renfro
>> What do you need int{ exp(sin(x)) dx } for?
Chris Rodgers wrote:
> It's for some quantum mechanics. The sin(x) arises
> because I apply an oscillating field which introduces
> sin(const * t) terms into the system energy levels.
> The exponential is for the quantum mechanical time
> evolution of a state in this system. It is very easy
> to calculate all this numerically, but not very
> informative on "why" things happen the way they do.
> I was hoping that a symbolic result might shed more
> light on things.
You may find the following book very helpful, and not just
with what you're currently interested in, but for other
things that are likely to come up as well.
"Advanced Mathematical Methods for Scientists and Engineers"
by Carl M. Bender and Steven A. Orszag
I had a course out of this book 23 years ago. If
someone or some company had approached me (or I had
known who to approach) to go into applied mathematics
and then do the kinds of things in this book for them
as a career, I might very well have followed a different
career path than I did. I loved the stuff in this book
(we only did about half of it in the one semester
graduate course I took) and I was constantly amazed
at how certain techniques were able to squeeze out
so much information in situations you'd think were
just hopeless. A lot of the exercises are old Putnam
Exam problems also, not that it helped me any -- my
Putnam days were over by then, such as it was. (Which
wasn't much of one. I think I was somewhere in the
upper 200's my Freshman year, and a lot worse than
that every year after that.)
I don't have the book with me now (but I do have
a copy at home), but I'd be surprised if the exact
function you're trying to analyze the antiderivative
behavior of isn't somewhere in this book. Probably
several places, because it seems to ring a bell to me.
Note that the antiderivative satisfies the differential
equation y' = (cos x)*y, which isn't all that exotic
looking to me (not that this should mean all that much).
One of the things I remember from this book is that
it's often easier to analyze asymptotic and qualitative
behavior by directly working with a differential
equation than with an integral or a series expression.
Dave L. Renfro
Robert Israel
Dave L. Renfro <renf...@cmich.edu> wrote:
>Dirk Van de moortel wrote (in part):
>
>>> What do you need int{ exp(sin(x)) dx } for?
>
>Chris Rodgers wrote:
>
>> It's for some quantum mechanics. The sin(x) arises
>> because I apply an oscillating field which introduces
>> sin(const * t) terms into the system energy levels.
>> The exponential is for the quantum mechanical time
>> evolution of a state in this system. It is very easy
>> to calculate all this numerically, but not very
>> informative on "why" things happen the way they do.
>> I was hoping that a symbolic result might shed more
>> light on things.
>I don't have the book with me now (but I do have
>a copy at home), but I'd be surprised if the exact
>function you're trying to analyze the antiderivative
>behavior of isn't somewhere in this book. Probably
>several places, because it seems to ring a bell to me.
>Note that the antiderivative satisfies the differential
>equation y' = (cos x)*y, which isn't all that exotic
>looking to me (not that this should mean all that much).
I think you mean, the integrand exp(sin(x)) satisfies
that DE. The antiderivative satisfies y'' = (cos x) y'.
It also satisfies
(y'' + y'''') (y')^2 + 2 (y'')^3 = 3 y' y'' y'''
(not that this is particularly helpful).
Dave L. Renfro
> I think you mean, the integrand exp(sin(x)) satisfies
> that DE. The antiderivative satisfies y'' = (cos x) y'.
> It also satisfies
> (y'' + y'''') (y')^2 + 2 (y'')^3 = 3 y' y'' y'''
> (not that this is particularly helpful).
Ooops! y' = (cos x)*y did seem almost too simple looking.
I guess this line of thought isn't going to get us
anywhere, even if I did know what to do when I got
there, which I don't. I'll try to remember to look
through the Bender/Orszag book later this evening
when I'm home. I'm pretty sure I've seen exp(sin x),
the function and not its antiderivative, somewhere
recently by the way (so unlikely to be Bender/Orszag,
although I feel sure it's there as well), probably
an old paper by J. W. L. Glashier or R. C. Archibald,
back when I was looking at the equation tan(x) = x
a year ago.
Dave L. Renfro
Robert Israel
G. A. Edgar <ed...@math.ohio-state.edu.invalid> wrote:
>In article <eo35b5$rhv$1...@frank-exchange-of-views.oucs.ox.ac.uk>, Chris
>Rodgers <rod...@physchem.NOSPAMox.aREMOVEc.uk> wrote:
>
>> / x=X
>> |
>> | exp(sin(x)) dx
>> |
>> / x=0
>>
>
>Certain definite integrals (X = 2 pi maybe) can be done
>in terms of Bessel functions.
int_0^{2 pi} exp(t sin(x)) dx = 2 pi I_0(t)
where I_0 is a modified Bessel function of the first kind.
Also
int_0^pi exp(t sin(x)) dx
= 2 int_0^{pi/2} exp(t sin(x)) dx
= pi (I_0(t) + L_0(t))
where L_0 is a modified Struve function.
I don't know about values of X that are not multiples of pi/2.
JEMebius
so myself (1:40 AM: the wee small hours).
Good luck: Johan E. Mebius
David W. Cantrell
> In article <100120071413457253%ed...@math.ohio-state.edu.invalid>,
> G. A. Edgar <ed...@math.ohio-state.edu.invalid> wrote:
> >In article <eo35b5$rhv$1...@frank-exchange-of-views.oucs.ox.ac.uk>, Chris
> >Rodgers <rod...@physchem.NOSPAMox.aREMOVEc.uk> wrote:
> >
> >> / x=X
> >> |
> >> | exp(sin(x)) dx
> >> |
> >> / x=0
> >>
> >
> >Certain definite integrals (X = 2 pi maybe) can be done
> >in terms of Bessel functions.
>
> int_0^{2 pi} exp(t sin(x)) dx = 2 pi I_0(t)
>
> where I_0 is a modified Bessel function of the first kind.
> Also
>
> int_0^pi exp(t sin(x)) dx
> = 2 int_0^{pi/2} exp(t sin(x)) dx
> = pi (I_0(t) + L_0(t))
>
> where L_0 is a modified Struve function.
I've found a symbolic result for the first integral which Chris wanted.
Expressed using Mathematica-style notation merely for easy comparison with
Bessel and Struve functions given at the Wolfram Functions site, my result
is that
Integrate[Exp[Sin[t]], {t, 0, x}]
is given by
| BesselI[0, 1] x + Pi/2 StruveL[0, 1] +
| 2 Sum[(-1)^k (BesselI[2k-1, 1] Cos[(2k-1)x]/(2k-1)
| + BesselI[2k, 1] Sin[2k x]/(2k)), {k, 1, Infinity}].
The Fourier series above converges reasonably rapidly. As a numerical
example, suppose we use x = 10 and take k only up to 3:
In[59]:= N[BesselI[0, 1] x + Pi/2 StruveL[0, 1] +
2 Sum[(-1)^k (BesselI[2k-1, 1] Cos[(2k-1) x]/(2k-1) +
BesselI[2k, 1] Sin[2k x]/(2k)), {k, 1, 3}] /. x -> 10, 10]
Out[59]= 14.60399071
Compare the above approximation with the more accurate
In[60]:= N[Integrate[Exp[Sin[x]], {x, 0, 10}], 10]
Out[60]= 14.60399098
David W. Cantrell
2*((-BesselI[1, 1])*
Cos[x] - BesselI[2, 1]*
(Sin[2*x]/2) +
BesselI[3, 1]*(Cos[3*x]/
3) + BesselI[4, 1]*
(Sin[4*x]/4) -
BesselI[5, 1]*(Cos[5*x]/
5) - BesselI[6, 1]*
(Sin[6*x]/6))