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On 5/12/21 5:53 PM, Peter Schoch wrote:
Do any of you know of a proof that shows the Method of Frobenius
could be used for anything other than just "regular singular point"
I tried re-doing all of my examples I gave in class using nothing but
Method of Frobenius, and they all worked!
That was exactly the right place to start. Straight out of Pólya.
Sometimes a special case is easier than the general case.
Sometimes vice versa.
However, examples are not a proof.
In this case not ... but they could have been, and you don't know
until you try. If any of the examples had failed, that would have
answered the student's question with a "no". It would have been
a solid proof (by construction) of the negative result. It likely
would have provided a fair bit of explanation.
Also note that googling this question would not have been easy.
You can find the wrong answer a lot more easily than the right
answer. In particular, the Frobenius wikipedia page speaks of:
"the requisite singularity at z = 0"
but we know it's not actually requisite. This wrong answer gets
repeated all over the place.
The trick is to keep googling until you find a /proof/ that it's
required ... or a /proof/ that it isn't. Proof by Bold Assertion
(PbBA) is not sufficient.
Here's further evidence that it's a nontrivial result: Frobenius
was not the village idiot. I guarantee you he asked himself this
question. Evidently he couldn't figure it out. It took Fuchs a
while to figure it out. So you shouldn't feel too bad if you
didn't find it to be immediately obvious.
Suggestion: Encourage the student to edit the wikipedia page.
Replace the wrong notion with the right notion, along with an
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