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Thank goodness for the internet - there is no way I could have finished this otherwise! Where could you go for reference if you are a non mathematician in the old days?
I have the expression and it help others who are stuck to google the "e" and the two cells just diagonally above it and to the right of it. Use the two letter word for the Greek symbol.

Good tip, Unclued.
Salisaturday, you might want to include the word 'Identity' in your search criteria.

It's a funny business though. If you play around for long enough with the square root of -1, particularly in a logarithmic way, then add 1 to it, you can end up with 0. To which everyone says "That's beautiful !"

I'm completely baffled. I have a completed grid, including the answers apparently too long, I have the number and both parts of the second quotation. I tried Kilgore Trout's advice of applying the number to successive rows and obtained complete gibberish apart from the last letters which spelt out the name of the author. I have little knowledge of mathematics whether pure or applied and don't know what to do next. What sort of "contradiction" should I be looking for? I certainly can't see one in the quotation.

If you look at the 2nd part of the 2nd quotation, there's a word that can be used as an anagram indicator. The gibberish should be an anagram of a key word in the 2nd half of the quote, contained in another part of the quote.

Escuan, if you have applied the key to the grid as Kilgore Trout suggested, after the first 6 letters, there are 11 letters which are an anagram of a word you mention in your post and then two letters which complete the first 6 letters. It's probably better to google the second quote (from the across clues) and you will find the entire quotation and why there is a contradiction.
However, I didn't find this terribly helpful in locating the "identity" (which is a mathematical term in this context). If you have completed the overlong solutions in the correct manner you should be able to find the "identity" by following Unclued's advice above.

I forgot to say that you have to unscramble the 11-letter anagram and change the letters in the grid to the correct word (as instructed in the preamble) before the "identity" will appear.

Could someone help me with 15D?

I've got the 1st 3 letters, of course, from the crossing clues (TAN_), and I suspect that it ends in an 'E', but have no idea how to parse the clue.

Thanks.

TANE is a form of ta'en = Shakespeare's (Bill's) taken, a word that can mean attracted. Marijuana is TEA with the E extra, and is 'N' and ecstasy is E.

I still don't understand what is meant by correcting the jumbled word in the grid. My jumble word begins AHS... and corrected it begins MAT. Does that mean I have to change the A and H in the 4th row to M and A etc. I don't see what that will achieve, but I know an e has to appear somewhere in the final grid.

"Bill" is Shakespeare, so it's a "Shakespearean" spelling of "taken" = "tane". Presumably it's "tea" (marijuana) '"n"(and) "e" (ecstasy) with the first "e" omitted. Other interpretations may be available.

Orson, you posted just before me! Yes, I presume you have to change all the letters of the "jumbled" word to get the "e" in the right place, even though it produces a load of non-words.