Calculate the polynomial coefficients of the Nth derivative of a polynomial coefficient wave

 Typically, people use Igor's Differentiate command to calculate derivatives. Suppose, then, that you are generating a curve from a set of polynomial coefficients W_coef; you typically first make a curve using poly(W_coef, x), and then call Differentiate. This approach has a number of downsides. It's susceptible to errors (which will depend on the density of points in your curve) and endpoint weirdness, not to mention that it uses unnecessary memory.

Alternatively, the snippet below shows how to calculate the coefficient wave corresponding to the Nth derivative of W_coef without ever using poly(W_coef, x). It returns this new coefficient wave as a reference to a free wave. You can make it global by using MoveWave or by changing the code to have /O instead of /FREE flags. Or, better yet, just embrace the freeness! (See Programming.ihf if you aren't used to Igor's new free waves.)

As an illustration of how convenient this is, consider the following, where W_coef has, say, 100 terms:

curve_15th_deriv = poly(PolyDeriv(W_coef, 15), x)


Boom! You just (analytically) calculated the 15th derivative of a 100-term polynomial in one line.

Function/WAVE PolyDeriv(coefs, n)       // Returns a (free) polynomial coefficient wave that is the nth derivative of the input polynomial coefficient wave
    Wave coefs      // polynomial coefficients
    Variable n      // positive integer >= 1
   
    if(n < numpnts(coefs))
        Make/FREE/D/N=(numpnts(coefs)-n) deriv
        deriv = (factorial(p+n)/factorial(p))*coefs[p+n]
    else
        Make/FREE/D/N=1 deriv = 0
    endif
   
    return deriv
End

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