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...
Posted: Wed Feb 24, 2010 11:38 am
 
Chip Eastham schrieb:
[quote]
On Feb 23, 2:54 pm, cliclic... at (no spam) freenet.de wrote:

If you merely want the antiderivative of #e^(x^b), that's a special
case of

int(x^(lambda-1) #e^(alpha x^p), x) =
= x^lambda/lambda #e^(alpha x^p) 1F1(1; 1+lambda/p; -alpha x^p)
[p /= 0, lambda/p /= 0, -1, -2, ...]
= -x^lambda/p #e^(alpha x^p) Psi(1, 1+lambda/p; -alpha x^p)
[p /= 0]

where 1F1(a;c;z) = Phi(a,c;z) and Psi(a,c;z) are the confluent
hypergeometric functions defined in G&R 9.210. It would surprise me
if Maple or Mathematica couldn't produce this result.


Thanks, Martin! Maxima could not do anything with it,

[/quote]
I should perhaps mention that, in spite of its style, the integral was
not lifted from *any* edition of Gradshteyn-Ryzhik, but taken from
personal notes of mine.

Martin.
 
Daniel Lichtblau...
Posted: Fri Feb 26, 2010 8:35 am
 
On Feb 26, 12:26 pm, cliclic... at (no spam) freenet.de wrote:
[quote]Robert Israel schrieb:





   lambda /        p                         lambda
  x       |(alpha x  p + lambda + p) GAMMA(- ------ + 1)
          \                                    p

              lambda + p                    lambda
        GAMMA(---------- + 1) binomial(1, - ------)
                  p                           p

                   lambda    2 p + lambda          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \          p
        (lambda + p) GAMMA(- ------ + ---------- + 1)| - alpha x  p
                               p          p          /

                lambda            lambda + p
        GAMMA(- ------ + 1) GAMMA(---------- + 1)
                  p                   p

                      lambda
        binomial(2, - ------)
                        p

                   lambda    lambda + 3 p          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \\
        (lambda + p) GAMMA(- ------ + ---------- + 1)||/lambda
                               p          p          //

Why does Maple 13 leave subexpressions like GAMMA(-lambda/p +
(lambda+p)/p + 1) unsimplified ?

And our elephant Mathematica is entirely helpless when asked to do
something with int(x^(lambda-1) #e^(alpha x^p), x) ?

Martin.
[/quote]
Well, you could check in Wolfram|Alpha.. I went to
http://www.wolframalpha.com
and entered
Integrate[x^(lambda-1)*Exp[alpha*x^p], x]

The first part of the resulting pane is copied below.

Indefinite integral:
integral x^(lambda-1) exp(alpha x^p) dx = -(x^lambda (alpha (-x^p))^(-
lambda/p) Gamma(lambda/p,-x^p alpha))/p+constant

I get a similar result when I use as input int(x^(lambda-1) e^(alpha
x^p), x).

Daniel Lichtblau
Wolfram Research
 
Daniel Lichtblau...
Posted: Fri Feb 26, 2010 11:04 am
 
On Feb 26, 1:59 pm, Axel Vogt <&nore... at (no spam) axelvogt.de> wrote:
[quote]Daniel Lichtblau wrote:
...

Well, you could check in Wolfram|Alpha [...]

This is very impressive (including the nice presentation of the
result [I only miss an additional ASCII output, directly and
not through properties of the image]).
[/quote]
I agree about desirability of a plain text format option. In fact I
looked for such a possibility before I did the cut/paste in my post.
I guess the issue is that many types of result are not amenable
to linear print formatting. But it strikes me as quite plausible to
check, and have a "plain text" (or maybe "Mathematica input
format" button to convert when feasible.


[quote]And what I really like (but use it rarely) with WolframAlpha:
it does *not* insist in a specific input syntax, it also takes
Maple syntax as input.

But gives up for the definite version int(..., x = 1 .. 2).
[/quote]
Yeah, it times out. Below is what I get on a fairly fast machine,
running Mathematica 7.0.1.

In[1]:= InputForm[Timing[Integrate[x^(lambda-1)*Exp[alpha*x^p], {x,
1,2}]]]

Out[1]//InputForm{11.819203, (ExpIntegralE[1 - lambda/p, -alpha] -
2^lambda*ExpIntegralE[1 - lambda/p, -(2^p*alpha)])/p}

That 12 seconds or so is more time than W|A will provide.

Daniel Lichtblau
Wolfram Research
 
Chip Eastham...
Posted: Fri Feb 26, 2010 11:56 am
 
On Feb 26, 1:35 pm, Daniel Lichtblau <d... at (no spam) wolfram.com> wrote:
[quote]On Feb 26, 12:26 pm, cliclic... at (no spam) freenet.de wrote:



Robert Israel schrieb:

   lambda /        p                         lambda
  x       |(alpha x  p + lambda + p) GAMMA(- ------ + 1)
          \                                    p

              lambda + p                    lambda
        GAMMA(---------- + 1) binomial(1, - ------)
                  p                           p

                   lambda    2 p + lambda          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \          p
        (lambda + p) GAMMA(- ------ + ---------- + 1)| - alpha x  p
                               p          p          /

                lambda            lambda + p
        GAMMA(- ------ + 1) GAMMA(---------- + 1)
                  p                   p

                      lambda
        binomial(2, - ------)
                        p

                   lambda    lambda + 3 p          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \\
        (lambda + p) GAMMA(- ------ + ---------- + 1)||/lambda
                               p          p          //

Why does Maple 13 leave subexpressions like GAMMA(-lambda/p +
(lambda+p)/p + 1) unsimplified ?

And our elephant Mathematica is entirely helpless when asked to do
something with int(x^(lambda-1) #e^(alpha x^p), x) ?

Martin.

Well, you could check in Wolfram|Alpha.. I went tohttp://www.wolframalpha..com
and entered
Integrate[x^(lambda-1)*Exp[alpha*x^p], x]

The first part of the resulting pane is copied below.

Indefinite integral:
integral x^(lambda-1) exp(alpha x^p) dx = -(x^lambda (alpha (-x^p))^(-
lambda/p) Gamma(lambda/p,-x^p alpha))/p+constant

I get a similar result when I use as input int(x^(lambda-1) e^(alpha
x^p), x).

Daniel Lichtblau
Wolfram Research
[/quote]
Thanks, Robert, Martin and Daniel. Wolfram
Alpha gives a fairly concise answer to:

Integrate[Exp(a*x^b),x]

(in terms of an incomplete gamma function)
but did not solve/simplify:

Integrate[Exp(a*x^b - x),x]

regards, chip
 
Daniel Lichtblau...
Posted: Fri Feb 26, 2010 12:17 pm
 
On Feb 26, 3:56 pm, Chip Eastham <hardm... at (no spam) gmail.com> wrote:
[quote]On Feb 26, 1:35 pm, Daniel Lichtblau <d... at (no spam) wolfram.com> wrote:



On Feb 26, 12:26 pm, cliclic... at (no spam) freenet.de wrote:

Robert Israel schrieb:

   lambda /        p                         lambda
  x       |(alpha x  p + lambda + p) GAMMA(- ------ + 1)
          \                                    p

              lambda + p                    lambda
        GAMMA(---------- + 1) binomial(1, - ------)
                  p                           p

                   lambda    2 p + lambda          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \          p
        (lambda + p) GAMMA(- ------ + ---------- + 1)| - alpha x  p
                               p          p          /

                lambda            lambda + p
        GAMMA(- ------ + 1) GAMMA(---------- + 1)
                  p                   p

                      lambda
        binomial(2, - ------)
                        p

                   lambda    lambda + 3 p          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \\
        (lambda + p) GAMMA(- ------ + ---------- + 1)||/lambda
                               p          p          //

Why does Maple 13 leave subexpressions like GAMMA(-lambda/p +
(lambda+p)/p + 1) unsimplified ?

And our elephant Mathematica is entirely helpless when asked to do
something with int(x^(lambda-1) #e^(alpha x^p), x) ?

Martin.

Well, you could check in Wolfram|Alpha.. I went tohttp://www.wolframalpha.com
and entered
Integrate[x^(lambda-1)*Exp[alpha*x^p], x]

The first part of the resulting pane is copied below.

Indefinite integral:
integral x^(lambda-1) exp(alpha x^p) dx = -(x^lambda (alpha (-x^p))^(-
lambda/p) Gamma(lambda/p,-x^p alpha))/p+constant

I get a similar result when I use as input int(x^(lambda-1) e^(alpha
x^p), x).

Daniel Lichtblau
Wolfram Research

Thanks, Robert, Martin and Daniel.  Wolfram
Alpha gives a fairly concise answer to:

Integrate[Exp(a*x^b),x]

(in terms of an incomplete gamma function)
but did not solve/simplify:

Integrate[Exp(a*x^b - x),x]

regards, chip
[/quote]
For specific rational values of b Mathematica can handle definite
integrals
of that sort, from 0 to infinity.

In[3]:= InputForm[Timing[ii = Integrate[Exp[a*x^(7/3)-x],
{x,0,Infinity}, Assumptions->a<0]]]

Out[3]//InputForm{5.7551250000000005, (-1680*3^(13/14)*(-
a)^(18/7)*Gamma[1/7]*Gamma[10/21]*
Gamma[17/21]*HypergeometricPFQ[{10/21, 17/21}, {2/7, 3/7, 4/7,
5/7, 6/7},
27/(823543*a^3)] - 630*3^(9/14)*(-
a)^(6/7)*Gamma[5/7]*Gamma[22/21]*
Gamma[29/21]*HypergeometricPFQ[{22/21, 29/21}, {6/7, 8/7, 9/7,
10/7,
11/7}, 27/(823543*a^3)] + 378*3^(1/14)*(-a)^(3/7)*Gamma[6/7]*
Gamma[25/21]*Gamma[32/21]*HypergeometricPFQ[{25/21, 32/21},
{8/7, 9/7, 10/7, 11/7, 12/7}, 27/(823543*a^3)] -
28*Pi*HypergeometricPFQ[{1, 4/3, 5/3}, {8/7, 9/7, 10/7, 11/7, 12/7,
13/7},
27/(823543*a^3)] -
630*(-1)^(6/7)*3^(5/14)*Sqrt[7]*a^(15/7)*Csc[Pi/7]*
Gamma[2/7]*Gamma[13/21]*Gamma[20/21]*HypergeometricPFQ[{13/21,
20/21},
{3/7, 4/7, 5/7, 6/7, 8/7}, 27/(823543*a^3)]*Sec[Pi/14]*Sec[(3*Pi)/
14] +
180*(-1)^(2/7)*3^(11/14)*Sqrt[7]*a^(12/7)*Csc[Pi/7]*Gamma[-4/7]*
Gamma[16/21]*Gamma[23/21]*HypergeometricPFQ[{16/21, 23/21},
{4/7, 5/7, 6/7, 8/7, 9/7}, 27/(823543*a^3)]*Sec[Pi/14]*Sec[(3*Pi)/
14] +
315*(-1)^(5/7)*3^(3/14)*Sqrt[7]*a^(9/7)*Csc[Pi/
7]*Gamma[4/7]*Gamma[19/21]*
Gamma[26/21]*HypergeometricPFQ[{19/21, 26/21}, {5/7, 6/7, 8/7,
9/7,
10/7}, 27/(823543*a^3)]*Sec[Pi/14]*Sec[(3*Pi)/14])/
(23520*a^3*Pi)}

Numerical sanity check:

In[5]:= InputForm[i2 = ii /. a->-3]
Out[5]//InputForm-(-45360*Sqrt[3]*Gamma[1/7]*Gamma[10/21]*Gamma[17/21]*
HypergeometricPFQ[{10/21, 17/21}, {2/7, 3/7, 4/7, 5/7, 6/7},
-1/823543] -
1890*Sqrt[3]*Gamma[5/7]*Gamma[22/21]*Gamma[29/21]*
HypergeometricPFQ[{22/21, 29/21}, {6/7, 8/7, 9/7, 10/7, 11/7},
-1/823543] + 378*Sqrt[3]*Gamma[6/7]*Gamma[25/21]*Gamma[32/21]*
HypergeometricPFQ[{25/21, 32/21}, {8/7, 9/7, 10/7, 11/7, 12/7},
-1/823543] - 28*Pi*HypergeometricPFQ[{1, 4/3, 5/3},
{8/7, 9/7, 10/7, 11/7, 12/7, 13/7}, -1/823543] +
5670*Sqrt[21]*Csc[Pi/7]*Gamma[2/7]*Gamma[13/21]*Gamma[20/21]*
HypergeometricPFQ[{13/21, 20/21}, {3/7, 4/7, 5/7, 6/7, 8/7},
-1/823543]*
Sec[Pi/14]*Sec[(3*Pi)/14] + 1620*Sqrt[21]*Csc[Pi/7]*Gamma[-4/7]*
Gamma[16/21]*Gamma[23/21]*HypergeometricPFQ[{16/21, 23/21},
{4/7, 5/7, 6/7, 8/7, 9/7}, -1/823543]*Sec[Pi/14]*Sec[(3*Pi)/14]
+
945*Sqrt[21]*Csc[Pi/7]*Gamma[4/7]*Gamma[19/21]*Gamma[26/21]*
HypergeometricPFQ[{19/21, 26/21}, {5/7, 6/7, 8/7, 9/7, 10/7},
-1/823543]*
Sec[Pi/14]*Sec[(3*Pi)/14])/(635040*Pi)

In[6]:= N[i2]
Out[6]= 0.407137

In[7]:= NIntegrate[Exp[-3*x^(7/3)-x], {x,0,Infinity}]
Out[7]= 0.407137

I would expect W|A to time out. I would not expect either to do
anything useful
when bounds are other than zero to infinity.

Getting back to the original topic, from which we all have
strayed...has anyone
yet managed to name this integral? (Would "Sheba" be appropriate?)

Daniel Lichtblau
Wolfram Research
 
Chip Eastham...
Posted: Fri Feb 26, 2010 1:13 pm
 
On Feb 26, 5:17 pm, Daniel Lichtblau <d... at (no spam) wolfram.com> wrote:
[quote]On Feb 26, 3:56 pm, Chip Eastham <hardm... at (no spam) gmail.com> wrote:



On Feb 26, 1:35 pm, Daniel Lichtblau <d... at (no spam) wolfram.com> wrote:

On Feb 26, 12:26 pm, cliclic... at (no spam) freenet.de wrote:

Robert Israel schrieb:

   lambda /        p                         lambda
  x       |(alpha x  p + lambda + p) GAMMA(- ------ + 1)
          \                                    p

              lambda + p                    lambda
        GAMMA(---------- + 1) binomial(1, - ------)
                  p                           p

                   lambda    2 p + lambda          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \          p
        (lambda + p) GAMMA(- ------ + ---------- + 1)| - alpha x  p
                               p          p          /

                lambda            lambda + p
        GAMMA(- ------ + 1) GAMMA(---------- + 1)
                  p                   p

                      lambda
        binomial(2, - ------)
                        p

                   lambda    lambda + 3 p          p    / /
        hypergeom([------], [------------], alpha x )  /  |
                     p            p                   /   \

                             lambda   lambda + p     \\
        (lambda + p) GAMMA(- ------ + ---------- + 1)||/lambda
                               p          p          //

Why does Maple 13 leave subexpressions like GAMMA(-lambda/p +
(lambda+p)/p + 1) unsimplified ?

And our elephant Mathematica is entirely helpless when asked to do
something with int(x^(lambda-1) #e^(alpha x^p), x) ?

Martin.

Well, you could check in Wolfram|Alpha.. I went tohttp://www.wolframalpha.com
and entered
Integrate[x^(lambda-1)*Exp[alpha*x^p], x]

The first part of the resulting pane is copied below.

Indefinite integral:
integral x^(lambda-1) exp(alpha x^p) dx = -(x^lambda (alpha (-x^p))^(-
lambda/p) Gamma(lambda/p,-x^p alpha))/p+constant

I get a similar result when I use as input int(x^(lambda-1) e^(alpha
x^p), x).

Daniel Lichtblau
Wolfram Research

Thanks, Robert, Martin and Daniel.  Wolfram
Alpha gives a fairly concise answer to:

Integrate[Exp(a*x^b),x]

(in terms of an incomplete gamma function)
but did not solve/simplify:

Integrate[Exp(a*x^b - x),x]

regards, chip

For specific rational values of b Mathematica can handle definite
integrals
of that sort, from 0 to infinity.

In[3]:= InputForm[Timing[ii = Integrate[Exp[a*x^(7/3)-x],
  {x,0,Infinity}, Assumptions->a<0]]]

Out[3]//InputForm> {5.7551250000000005, (-1680*3^(13/14)*(-
a)^(18/7)*Gamma[1/7]*Gamma[10/21]*
    Gamma[17/21]*HypergeometricPFQ[{10/21, 17/21}, {2/7, 3/7, 4/7,
5/7, 6/7},
     27/(823543*a^3)] - 630*3^(9/14)*(-
a)^(6/7)*Gamma[5/7]*Gamma[22/21]*
    Gamma[29/21]*HypergeometricPFQ[{22/21, 29/21}, {6/7, 8/7, 9/7,
10/7,
      11/7}, 27/(823543*a^3)] + 378*3^(1/14)*(-a)^(3/7)*Gamma[6/7]*
    Gamma[25/21]*Gamma[32/21]*HypergeometricPFQ[{25/21, 32/21},
     {8/7, 9/7, 10/7, 11/7, 12/7}, 27/(823543*a^3)] -
   28*Pi*HypergeometricPFQ[{1, 4/3, 5/3}, {8/7, 9/7, 10/7, 11/7, 12/7,
13/7},
     27/(823543*a^3)] -
630*(-1)^(6/7)*3^(5/14)*Sqrt[7]*a^(15/7)*Csc[Pi/7]*
    Gamma[2/7]*Gamma[13/21]*Gamma[20/21]*HypergeometricPFQ[{13/21,
20/21},
     {3/7, 4/7, 5/7, 6/7, 8/7}, 27/(823543*a^3)]*Sec[Pi/14]*Sec[(3*Pi)/
14] +
   180*(-1)^(2/7)*3^(11/14)*Sqrt[7]*a^(12/7)*Csc[Pi/7]*Gamma[-4/7]*
    Gamma[16/21]*Gamma[23/21]*HypergeometricPFQ[{16/21, 23/21},
     {4/7, 5/7, 6/7, 8/7, 9/7}, 27/(823543*a^3)]*Sec[Pi/14]*Sec[(3*Pi)/
14] +
   315*(-1)^(5/7)*3^(3/14)*Sqrt[7]*a^(9/7)*Csc[Pi/
7]*Gamma[4/7]*Gamma[19/21]*
    Gamma[26/21]*HypergeometricPFQ[{19/21, 26/21}, {5/7, 6/7, 8/7,
9/7,
      10/7}, 27/(823543*a^3)]*Sec[Pi/14]*Sec[(3*Pi)/14])/
(23520*a^3*Pi)}

Numerical sanity check:

In[5]:= InputForm[i2 = ii /. a->-3]
Out[5]//InputForm> -(-45360*Sqrt[3]*Gamma[1/7]*Gamma[10/21]*Gamma[17/21]*
    HypergeometricPFQ[{10/21, 17/21}, {2/7, 3/7, 4/7, 5/7, 6/7},
-1/823543] -
   1890*Sqrt[3]*Gamma[5/7]*Gamma[22/21]*Gamma[29/21]*
    HypergeometricPFQ[{22/21, 29/21}, {6/7, 8/7, 9/7, 10/7, 11/7},
     -1/823543] + 378*Sqrt[3]*Gamma[6/7]*Gamma[25/21]*Gamma[32/21]*
    HypergeometricPFQ[{25/21, 32/21}, {8/7, 9/7, 10/7, 11/7, 12/7},
     -1/823543] - 28*Pi*HypergeometricPFQ[{1, 4/3, 5/3},
     {8/7, 9/7, 10/7, 11/7, 12/7, 13/7}, -1/823543] +
   5670*Sqrt[21]*Csc[Pi/7]*Gamma[2/7]*Gamma[13/21]*Gamma[20/21]*
    HypergeometricPFQ[{13/21, 20/21}, {3/7, 4/7, 5/7, 6/7, 8/7},
-1/823543]*
    Sec[Pi/14]*Sec[(3*Pi)/14] + 1620*Sqrt[21]*Csc[Pi/7]*Gamma[-4/7]*
    Gamma[16/21]*Gamma[23/21]*HypergeometricPFQ[{16/21, 23/21},
     {4/7, 5/7, 6/7, 8/7, 9/7}, -1/823543]*Sec[Pi/14]*Sec[(3*Pi)/14]
+
   945*Sqrt[21]*Csc[Pi/7]*Gamma[4/7]*Gamma[19/21]*Gamma[26/21]*
    HypergeometricPFQ[{19/21, 26/21}, {5/7, 6/7, 8/7, 9/7, 10/7},
-1/823543]*
    Sec[Pi/14]*Sec[(3*Pi)/14])/(635040*Pi)

In[6]:= N[i2]
Out[6]= 0.407137

In[7]:= NIntegrate[Exp[-3*x^(7/3)-x], {x,0,Infinity}]
Out[7]= 0.407137

I would expect W|A to time out. I would not expect either to do
anything useful
when bounds are other than zero to infinity.

Getting back to the original topic, from which we all have
strayed...has anyone
yet managed to name this integral? (Would "Sheba" be appropriate?)

Daniel Lichtblau
Wolfram Research
[/quote]
I meant the subject line somewhat tongue-
in-cheek, like "name that song". But Sheba
works for me!

I suspect we will be seeing a post from the
originator (it has some physical application)
who posed it in terms of 8 or 9 parameters.
Mathematically it comes down to something
like the form above, and since for b = 2,
we have something like the error function,
I've advised him we cannot expect anything
in terms of elementary transcendentals.

I'm not clear about what the limits of
integration are supposed to be. I actually
think a numerical integration would prove
completely satisfactory, but many times an
"exact" solution is what people ask for.

regards, chip
 
...
Posted: Fri Feb 26, 2010 1:26 pm
 
Robert Israel schrieb:
[quote]
lambda / p lambda
x |(alpha x p + lambda + p) GAMMA(- ------ + 1)
\ p

lambda + p lambda
GAMMA(---------- + 1) binomial(1, - ------)
p p

lambda 2 p + lambda p / /
hypergeom([------], [------------], alpha x ) / |
p p / \

lambda lambda + p \ p
(lambda + p) GAMMA(- ------ + ---------- + 1)| - alpha x p
p p /

lambda lambda + p
GAMMA(- ------ + 1) GAMMA(---------- + 1)
p p

lambda
binomial(2, - ------)
p

lambda lambda + 3 p p / /
hypergeom([------], [------------], alpha x ) / |
p p / \

lambda lambda + p \\
(lambda + p) GAMMA(- ------ + ---------- + 1)||/lambda
p p //

[/quote]
Why does Maple 13 leave subexpressions like GAMMA(-lambda/p +
(lambda+p)/p + 1) unsimplified ?

And our elephant Mathematica is entirely helpless when asked to do
something with int(x^(lambda-1) #e^(alpha x^p), x) ?

Martin.
 
Axel Vogt...
Posted: Fri Feb 26, 2010 2:59 pm
 
Daniel Lichtblau wrote:
....
[quote]
Well, you could check in Wolfram|Alpha.. I went to
http://www.wolframalpha.com
and entered
Integrate[x^(lambda-1)*Exp[alpha*x^p], x]

The first part of the resulting pane is copied below.

Indefinite integral:
integral x^(lambda-1) exp(alpha x^p) dx = -(x^lambda (alpha (-x^p))^(-
lambda/p) Gamma(lambda/p,-x^p alpha))/p+constant

I get a similar result when I use as input int(x^(lambda-1) e^(alpha
x^p), x).

Daniel Lichtblau
Wolfram Research
[/quote]
This is very impressive (including the nice presentation of the
result [I only miss an additional ASCII output, directly and
not through properties of the image]).

And what I really like (but use it rarely) with WolframAlpha:
it does *not* insist in a specific input syntax, it also takes
Maple syntax as input.

But gives up for the definite version int(..., x = 1 .. 2).
 
mark mcclure...
Posted: Fri Feb 26, 2010 3:15 pm
 
On Feb 26, 2:59 pm, Axel Vogt <&nore... at (no spam) axelvogt.de> wrote:

[quote]And what I really like (but use it rarely) with WolframAlpha:
it does *not* insist in a specific input syntax, it also takes
Maple syntax as input.
[/quote]
I have strongly mixed feelings about WolframAlpha. While I like its
ease of use and suspect that students will use it for exactly that
reason, its "syntax" is a bit too loose I think. I don't think I
would say that WolframAlpha understands Maple syntax, or even
Mathematica syntax. There are plenty of examples of syntactically
correct Mathematica expressions that WolframAlpha interprets
differently. One example that is tangentially related to this
discussion is Integrate[1/2x,x]. Mathematica correctly interprets this
to be Integrate[(1/2)x, x], while WolframAlpha interprets it to be
Integrate[1/(2x), x].

Of course, WolframAlpha is not case sensitive, you can generally swap
brackets and parentheses, and it doesn't recognize the options of any
Mathematica commands that I've tried. To the extent that it
understands commands from any system at all, I think, is reflective of
the fact that most commands in any CAS were chosen in a sensible way.

Mark McClure
 
A N Niel...
Posted: Fri Feb 26, 2010 3:27 pm
 
In article <4B881248.36C567D5 at (no spam) freenet.de>, <clicliclic at (no spam) freenet.de>
wrote:
[quote]
Why does Maple 13 leave subexpressions like GAMMA(-lambda/p +
(lambda+p)/p + 1) unsimplified ?
[/quote]
Because it was not told to simplify?

GAMMA(-lambda/p +(lambda+p)/p + 1);
/ lambda lambda + p \
GAMMA|- ------ + ---------- + 1|
\ p p /
simplify(%);
1
 
Axel Vogt...
Posted: Fri Feb 26, 2010 4:18 pm
 
Daniel Lichtblau wrote:
[quote]On Feb 26, 1:59 pm, Axel Vogt <&nore... at (no spam) axelvogt.de> wrote:
Daniel Lichtblau wrote:
...

Well, you could check in Wolfram|Alpha [...]
This is very impressive (including the nice presentation of the
result [I only miss an additional ASCII output, directly and
not through properties of the image]).

I agree about desirability of a plain text format option. In fact I
looked for such a possibility before I did the cut/paste in my post.
I guess the issue is that many types of result are not amenable
to linear print formatting. But it strikes me as quite plausible to
check, and have a "plain text" (or maybe "Mathematica input
format" button to convert when feasible.
[/quote]
Actually your system writes to the image and I can copy from
its properties :-)

Anyway: it is much more 'friendly' than what MMA provided through
its Integrator at http://integrals.wolfram.com/index.jsp

[quote]

And what I really like (but use it rarely) with WolframAlpha:
it does *not* insist in a specific input syntax, it also takes
Maple syntax as input.

But gives up for the definite version int(..., x = 1 .. 2).

Yeah, it times out. Below is what I get on a fairly fast machine,
running Mathematica 7.0.1.

In[1]:= InputForm[Timing[Integrate[x^(lambda-1)*Exp[alpha*x^p], {x,
1,2}]]]

Out[1]//InputForm=
{11.819203, (ExpIntegralE[1 - lambda/p, -alpha] -
2^lambda*ExpIntegralE[1 - lambda/p, -(2^p*alpha)])/p}

That 12 seconds or so is more time than W|A will provide.
[/quote]
Does it need the time to check, whether it can apply
solution(upperBound) - solution(lowerBound) by looking
for continuity? Or else (using ExpIntegralE instead of
incomplete Gamma)?
 
 
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