User uri cohen - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:26:28Z http://mathoverflow.net/feeds/user/10450 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115272/taylor-expansion-convergence-relation-to-power-spectrum Taylor expansion convergence relation to power-spectrum Uri Cohen 2012-12-03T11:16:43Z 2012-12-03T14:34:23Z <p>Is there some connection between the power-spectrum of a real function $f:\mathbb{R}\to\mathbb{R}$ (that is, its Fourier transform) and the convergence radius of its Taylor expansion around arbitraty $x_0$? Intuitively, I would expect a function with 'limited power at high frequencies' to have 'large convergence radius' around each point, but I could not find such result.</p> <p>Thanks!</p> http://mathoverflow.net/questions/12828/inverse-gamma-function/98267#98267 Answer by Uri Cohen for Inverse gamma function? Uri Cohen 2012-05-29T11:54:13Z 2012-05-29T11:54:13Z <p>For the benefit of generations to come I add here the python code I wrote after reading the above answers.</p> <pre><code>import numpy as np import math import scipy.special def _lambert_w(z): """ Lambert W function, principal branch. See http://en.wikipedia.org/wiki/Lambert_W_function Code taken from http://keithbriggs.info/software.html """ eps=4.0e-16 em1=0.3678794411714423215955237701614608 assert z&gt;=-em1, 'LambertW.py: bad argument %g, exiting.'%z if 0.0==z: return 0.0 if z&lt;-em1+1e-4: q=z+em1 r=math.sqrt(q) q2=q*q q3=q2*q return\ -1.0\ +2.331643981597124203363536062168*r\ -1.812187885639363490240191647568*q\ +1.936631114492359755363277457668*r*q\ -2.353551201881614516821543561516*q2\ +3.066858901050631912893148922704*r*q2\ -4.175335600258177138854984177460*q3\ +5.858023729874774148815053846119*r*q3\ -8.401032217523977370984161688514*q3*q if z&lt;1.0: p=math.sqrt(2.0*(2.7182818284590452353602874713526625*z+1.0)) w=-1.0+p*(1.0+p*(-0.333333333333333333333+p*0.152777777777777777777777)) else: w=math.log(z) if z&gt;3.0: w-=math.log(w) for i in xrange(10): e=math.exp(w) t=w*e-z p=w+1.0 t/=e*p-0.5*(p+1.0)*t/p w-=t if abs(t)&lt;eps*(1.0+abs(w)): return w raise AssertionError, 'Unhandled value %1.2f'%z def _gamma_inverse(x): """ Inverse the gamma function. http://mathoverflow.net/questions/12828/inverse-gamma-function """ k=1.461632 # the positive zero of the digamma function, scipy.special.psi assert x&gt;=k, 'gamma(x) is strictly increasing for x &gt;= k, k=%1.2f, x=%1.2f' % (k, x) C=math.sqrt(2*np.pi)/np.e - scipy.special.gamma(k) # approximately 0.036534 L=np.log((x+C)/np.sqrt(2*np.pi)) gamma_inv = 0.5+L/_lambert_w(L/np.e) return gamma_inv </code></pre> http://mathoverflow.net/questions/115272/taylor-expansion-convergence-relation-to-power-spectrum/115274#115274 Comment by Uri Cohen Uri Cohen 2012-12-03T12:49:46Z 2012-12-03T12:49:46Z Nice. Can you provide more details on this? Maybe a link or a name for this phenomenon which I can follow?