06-1350/Some Equations by Kurlin - Revision history

Drorbn: Reverted edit of 164.58.61.52, changed back to last version by Drorbn

2007-06-08T10:12:49Z

Reverted edit of 164.58.61.52, changed back to last version by Drorbn

← Older revision		Revision as of 06:12, 8 June 2007
Line 4:		Line 4:

	<center>		<center>
	<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

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	<center>		<center>
	<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

164.58.61.52 at 05:37, 8 June 2007

2007-06-08T05:37:57Z

← Older revision		Revision as of 01:37, 8 June 2007
Line 4:		Line 4:

	<center>		<center>
	<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

Drorbn: Reverted edit of 68.38.99.126, changed back to last version by Drorbn

2007-05-28T14:50:25Z

Reverted edit of 68.38.99.126, changed back to last version by Drorbn

← Older revision		Revision as of 10:50, 28 May 2007
Line 4:		Line 4:

	<center>		<center>
	<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

68.38.99.126 at 06:29, 28 May 2007

2007-05-28T06:29:46Z

← Older revision		Revision as of 02:29, 28 May 2007
Line 4:		Line 4:

	<center>		<center>
	<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

Drorbn at 01:11, 10 May 2007

2007-05-10T01:11:36Z

← Older revision		Revision as of 21:11, 9 May 2007
Line 1:		Line 1:
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	'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by		'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by

	<center>		<center>
	<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

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	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

133.11.55.34 at 22:16, 9 May 2007

2007-05-09T22:16:45Z

← Older revision		Revision as of 18:16, 9 May 2007
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	'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by		'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by

	<center>		<center>
	<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

Drorbn: Reverted edit of 58.188.46.157, changed back to last version by Drorbn

2007-05-08T12:15:52Z

Reverted edit of 58.188.46.157, changed back to last version by Drorbn

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	'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by		'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by

	<center>		<center>
	<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

58.188.46.157 at 06:32, 8 May 2007

2007-05-08T06:32:00Z

← Older revision		Revision as of 02:32, 8 May 2007
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	'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by		'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by

	<center>		<center>
	<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

Drorbn: Reverted edit of 216.147.215.194, changed back to last version by Drorbn

2007-04-30T10:14:51Z

Reverted edit of 216.147.215.194, changed back to last version by Drorbn

← Older revision		Revision as of 06:14, 30 April 2007
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	'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by		'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by

	<center>		<center>
	<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?

216.147.215.194 at 07:33, 30 April 2007

2007-04-30T07:33:13Z

← Older revision		Revision as of 03:33, 30 April 2007
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	{{06-1350/Navigation}}

	'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by		'''Claim''' {{ref\|Kurlin}}. The function <math>f(\lambda,\mu)</math> defined by

	<center>		<center>
	<math>1+2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda+\mu}-e^{-\lambda-\mu}}{2(\lambda+\mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}}+\frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>		<math>1 2\lambda\mu f(\lambda,\mu) = \frac{e^{\lambda \mu}-e^{-\lambda-\mu}}{2(\lambda \mu)} \left(\frac{2\lambda}{e^\lambda-e^{-\lambda}} \frac{2\mu}{e^\mu-e^{-\mu}}-1\right)</math>
	</center>		</center>

Line 10:		Line 10:

	<center>		<center>
	<math>f(\lambda,\mu)+e^\mu f(\mu,-\lambda-\mu)+e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda+\mu} \left(\frac{e^\mu-1}{\mu}+\frac{e^{-\lambda}-1}{\lambda}\right).</math>		<math>f(\lambda,\mu) e^\mu f(\mu,-\lambda-\mu) e^{-\lambda}f(\lambda,-\lambda-\mu) = \frac{1}{\lambda \mu} \left(\frac{e^\mu-1}{\mu} \frac{e^{-\lambda}-1}{\lambda}\right).</math>
	</center>		</center>

Line 20:		Line 20:

	* Is that always the case?		* Is that always the case?
	** A pretty example is the tame equation <math>e(x+y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.		** A pretty example is the tame equation <math>e(x y)=e(x)e(y)</math>, whose solution is the "first" transcendental function <math>e^x</math>.
	** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?		** Are we done defining "new" special functions after we've defined <math>e^x</math> or are there further ones that cannot be defined in terms of it?
	** If there are further ones, why haven't we heard about them? Or else, where do we read about them?		** If there are further ones, why haven't we heard about them? Or else, where do we read about them?