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File:Illustration of Daffodil Lemma from Analytic Combinatorics pp 267.png

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Summary

Description
English: Polar plot of for . This illustrates the 'Daffodil lemma', relating the periodic fluctuations of the coefficients of a generating function with the location of its dominant singularities. Reproduction of a graph from Flajolet, Philippe; Sedgewick, Robert (2009) Analytic Combinatorics, Cambridge University Press pp. 390.
Date
Source Own work
Author Dom walden
PNG development
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This plot was created with Matplotlib.
Source code
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Python code

import matplotlib.pyplot as plt
import numpy as np

def f(z):
    return z**7 * np.e**(z**25) + z**2 / (1 - z**10)

fig, ax = plt.subplots(subplot_kw={'projection': 'polar'})

theta = np.arange(0, 2 * np.pi, 0.01)
ax.plot(theta, abs(f(0.4*np.e**(1j*theta))), label='0.4')
ax.plot(theta, abs(f(0.5*np.e**(1j*theta))), label='0.5')
ax.plot(theta, abs(f(0.6*np.e**(1j*theta))), label='0.6')
ax.plot(theta, abs(f(0.7*np.e**(1j*theta))), label='0.7')
ax.plot(theta, abs(f(0.8*np.e**(1j*theta))), label='0.8')
ax.plot(theta, abs(f(0.9*np.e**(1j*theta))), label='0.9')

ax.set_rmax(2)
ax.set_rticks([0, 0.5, 1, 1.5, 2])
ax.set_rlabel_position(0)

ax.set_xticks([0, np.pi/2, np.pi, 3*np.pi/2], labels=['', '', '', ''])

ax.grid(True)

ax.legend()
ax.set_title('Polar plot of $|f(re^{i\\theta})|$ for $f(z) = z^7 e^{z^{25} } + \\frac{z^2}{(1 - z^{10})}$, $r = 0.4..0.9$', va='bottom')
plt.show()

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Captions

Polar plot illustrating the Daffodil Lemma. Reproduction of a graph from Flajolet and Sedgewick "Analytic Combinatorics" pp. 267.

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inception

9 March 2025

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current11:27, 9 March 2025Thumbnail for version as of 11:27, 9 March 20251,596 × 810 (135 KB)Dom waldenUploaded own work with UploadWizard

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