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Hi everybody. Not owning any HP48/49 model, there's no immediate way for me to test this, so I'd appreciate if any of you HP48/49 owners would try this for me and post here the results.
The question is: what result does the HP48/49 symbolic algebra system produce for the following indefinite integral:
Integral[ sin(x^n) . dx ]
where n is a constant ? Specifically, I'm interested to know if it does produce some kind of exact closed result, in terms of known functions (elementary or not). I'm not interested in Taylorseriesexpansion based results or any other nonclosed approximations (not to mention purely numerical results).
In case it can't produce a closed result for general, arbitrary n, I would then be interested to know if it can produce a closed result for some particular values of n, and if so, the specific values of n it can solve. I would expect it to be able to produce a closed result for n = 0 and n = 1 at the very least, but what about other values of n, both integer and noninteger (n = 2, 3, ..., 1/2, 1/3, ...) ? What about negative values of n (n = 1, 2, ..., 1/2, ...) ?
Thanks in advance for any results or comments and best regards from V.
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Hi Valentin.
Firstly, the 48 does not dispose of a CAS worth mentioning;
And the 49 cannot solve any but the most trivial of these integrals (ie with N=0 or 1). All the rest are simply
returned in integral form. Not a great surprise, as
for N=2 the result is a Fresnel integral, and for larger
n it is expressible as complex incomplete gamma functions,
both of which are not builtin to the 49G.
Cheers, Werner
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Werner posted:
"The 49 cannot solve any but the most trivial of these integrals (ie with N=0 or 1). All the rest are simply returned in integral form."
Thanks for replying, Werner, but have you actually tried it on an HP49, or is your comment just what you think based on your experience ? I would need an actual attempt on the calculator.
"Not a great surprise, as for N=2 the result is a Fresnel integral, and for larger n it is expressible as
complex incomplete gamma functions, both of which are not builtin to the 49G."
But what about noninteger and/or negative values of N ? Have you actually tried some ? Which ? If it cannot deal
with values other than N=0 and N=1, it *would* be somewhat of a surprise, at least for me ... Please try it on an actual machine, if you haven't done already so, and tell me what values did you try and what results did it produce.
Best regards from V.
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Valentín:
I have used an HP49G emulator downloaded from hpcalc.org and I got the following answers:
INTVX(SIN(X^N)) = INT(SIN(EXP(N*LN(X)))
RISCH(SIN(X^N),X) = INT(SIN(EXP(N*LN(X)))
I did no try the IBP (integration by parts) command because the function to be integrated must be expressed as a product of functions.
The ROM in the emulator was 1.18
It seems that this is too much for the HP49G CAS.
Rolando
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you're expecting a bit much arent you. these integratals are mostly only in terms of special functions (like complex gamma, sinc, and fresnel) which arent even there. except for the cases n = 0, 1 and 1/2.
interstingly the case n = 1, is close to one of my numerical tests: integrate(cos(1/x), 0, 1) numerically or otherwise. so far no calculator has even got close (0.0844109505)
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there's some more. 1/m (m = 1,2...) are also representable since int(sin(x^n), dx) can be solved when int(x^n sin(x), dx) is soluble.
i'd be interested to know if these come out of the 48/49, using alg48, for example.
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Hi Valentin. I did use a real 49G in the process:
n>1 or negative: integral returned
n=0,1 OK
n=1/2 '2*SIN(SQRT(X))2*SQRT(X)*COS(SQRT(X))'
n=1/2,n=1/3 integral returned
That was in 'real mode'.
In complex mode, nothing much changes.
For negative n, the integral is split in two parts that
are both left unintegrated, eg for n=1 I get:
'INT((i*EXP(i/Xt)/2),Xt,X)+INT(i/(2*EXP(i/Xt)),Xt,X)'
The Xt stuff is a change of variable and the 4x's way of saying it can't solve the integral.
Cheers, Werner
BTW you can play with a 49 yourself using the excellent Emu48 emulator and a 49G ROM, both of which can be found
at www.hpcalc.org
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Thanks Werner, it's more or less what I expected and what would be reasonable, by the way. The fact that it manages to solve the n=1/2 case is encouraging.
I've always been somewhat amazed by the fact that sin(sqrt(x)) is elementarily integrable while sin(x^2) is not. One would tend to expect exactly the opposite, as the square root function seems more 'irrational', so to speak, than the simple, polynomial, 'square' function :)
Best regards from V.
