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From: percus@acf4.UUCP (Allon G. Percus)
Newsgroups: net.math
Subject: Re: value of an integral
Message-ID: <5980010@acf4.UUCP>
Date: Fri, 21-Feb-86 19:25:00 EST
Article-I.D.: acf4.5980010
Posted: Fri Feb 21 19:25:00 1986
Date-Received: Mon, 24-Feb-86 07:20:39 EST
References: <823@drux2.UUCP>
Organization: New York University
Lines: 96

> However, techniques as used to find the value of:
> 
> int from 0 to inf e sup x sup 2 dx
> 
> are acceptable.

I'm afraid to say it, but as it happens,

				   inf
				  /	 2
				  [     x
 				  I    e   dx
				  ]
				  /
				   0

diverges.

What you want is:

				   inf
				  /	   2
				  [     - x
 				  I    e     dx
				  ]
				  /
				   0

To solve, observe that this is just:

			 |-----------------------------------
			 |     inf	       inf
			 |    /	       2      /	       2
			 |    [	    - x	      [	    - y
			 |    I	   e     dx   I    e     dy
			 |    ]		      ]
			 |    /		      /
			\|     0	       0
Which is:
			 |-----------------------------------
			 |     inf    inf
			 |    /      /	      2	   2
			 |    [	     [	   - x  - y
			 |    I	     I    e	     dx dy
			 |    ]	     ]
			 |    /	     /
			\|     0      0

Now convert this to polar coordinates:

			 |-----------------------------------
			 |     pi/2   inf
			 |    /      /	      2
			 |    [	     [	   - r
			 |    I	     I    e     r dr dtheta
			 |    ]	     ]
			 |    /	     /
			\|     0      0
Which is:
			 |-----------------------------------
			 |     pi/2  [		] inf
			 |    /      |	     2	|
			 |    [	     |	  - r	|
			 |    I	     |   e	|     dtheta
			 |    ]	     |   -----	|
			 |    /	     |	  - 2 	|
			\|     0     [		] 0

Or:
				 |-----------------------
				 |     pi/2
				 |    /
				 |    [	     1
				 |    I	     -  dtheta
				 |    ]	     2
				 |    /
				\|     0



Which finally becomes:

					sqrt(pi)
					--------
					    2

Using a similar technique, you should be able to solve your problem.

           .
        -------
        |-----|             A. G. Percus
        |II II|      (ARPA) percus@acf4
        |II II|       (NYU) percus.acf4
        |II II|      (UUCP) ...{allegra!ihnp4!seismo}!cmcl2!acf4!percus
        |II II|
        -------