Can it be used for higher-order catastrophe integrals?

[Malibiea]

The program is super great! I would like to know if extending the phase coefficients can make z exponentially -2. Or it can realize double integrals, which are useful for higher-order catastrophe integrals， such as elliptic umbilic canonical integrals and hyperbolic umbilic canonical integrals.

[AndrewGibbs]

Thank you for the kind words!

I will answer each question one by one:

• I'm not sure what you mean by "make z exponentially -2"? Can you explain?
• Double integrals are not something we are thinking about currently - it's too hard. You can of course evaluate integrals with separable phase g(x,y) = X(x) + Y(y) easily already - but I expect what you have in mind is trickier than this. You could try using this code.
• We are working on an extension to rational phase functions, which could be applied to the single integral form of the Umbrillic catastrophes. This is quite tricky, though, and it will probably not be for another year at least... so don't wait on us!

Andrew

[Malibiea]

Dear Andrew,

• I'm sorry I didn't describe the first question well. The phase function I need to solve might be in the form shown in the picture. I don't know how to input this phase function into the program, because of the third term. The exponent of x is negative 2. As if it couldn't be implemented via polyCoeffs in the program. I have not yet tried whether the third term in the phase function can be implemented by building a custom function in f. If this project can extend the functionality of negative exponents or more complex phase functions, it will have a wider application in related fields.

• Yeah, it's true. The high-order catastrophe integrals are pretty tricky. Thank you for your suggestions.

Best, Malibiea