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Sidharth Ghoshal
Sidharth Ghoshal
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There was a question asking about extending the partition function to a continuous function on math.stackexchange a while ago. While there are infinitely many answers to this I believe "JM Ain't a Mathematician" was able to single out a particular sequence of such functions he felt was natural and in the limit of this sequence there appears to be a unique convex solution. See here: https://math.stackexchange.com/questions/34318/feeding-real-or-even-complex-numbers-to-the-integer-partition-function-pn

Now if you want to invert the partition function, what you could do is try to evaluate that limit function in the link as a taylor series and then apply the lagrange inversion theorem to get your inverse partition function.

You might even be able to end up with a relation on the entire complex plane this way.

There was a question asking about extending the partition function to a continuous function on math.stackexchange a while ago. While there are infinitely many answers to this I believe "JM Ain't a Mathematician" was able to single out a particular sequence of such functions he felt was natural and in the limit of this sequence there appears to be a unique convex solution. See here: https://math.stackexchange.com/questions/34318/feeding-real-or-even-complex-numbers-to-the-integer-partition-function-pn

Now if you want to invert the partition function, what you could do is try to evaluate that limit function in the link as a taylor series and then apply the lagrange inversion theorem to get your inverse partition function.

There was a question asking about extending the partition function to a continuous function on math.stackexchange a while ago. While there are infinitely many answers to this I believe "JM Ain't a Mathematician" was able to single out a particular sequence of such functions he felt was natural and in the limit of this sequence there appears to be a unique convex solution. See here: https://math.stackexchange.com/questions/34318/feeding-real-or-even-complex-numbers-to-the-integer-partition-function-pn

Now if you want to invert the partition function, what you could do is try to evaluate that limit function in the link as a taylor series and then apply the lagrange inversion theorem to get your inverse partition function.

You might even be able to end up with a relation on the entire complex plane this way.

Source Link
Sidharth Ghoshal
Sidharth Ghoshal
  • 2.7k
  • 19
  • 32

There was a question asking about extending the partition function to a continuous function on math.stackexchange a while ago. While there are infinitely many answers to this I believe "JM Ain't a Mathematician" was able to single out a particular sequence of such functions he felt was natural and in the limit of this sequence there appears to be a unique convex solution. See here: https://math.stackexchange.com/questions/34318/feeding-real-or-even-complex-numbers-to-the-integer-partition-function-pn

Now if you want to invert the partition function, what you could do is try to evaluate that limit function in the link as a taylor series and then apply the lagrange inversion theorem to get your inverse partition function.