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import numpy as np
# The Lobachevsky function is defined as:
# Λ(θ) = 0.5 * Σ(k=1 to ∞) sin(2kθ)/k²
# Its derivative is:
# dΛ/dθ = 0.5 * Σ(k=1 to ∞) 2k*cos(2kθ)/k² = Σ(k=1 to ∞) cos(2kθ)/k
# Let's verify this matches the claimed formula log(2|sin(θ)|)
def lob_series(theta, n=100):
"""Compute Lobachevsky function using series"""
result = 0.0
for k in range(1, n+1):
result += np.sin(2*k*theta) / (k*k)
return 0.5 * result
def lob_deriv_series(theta, n=100):
"""Compute derivative using series"""
result = 0.0
for k in range(1, n+1):
result += np.cos(2*k*theta) / k
return result
# Test at several angles
angles = [0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0]
print("Comparing derivative formulas:")
print("θ Series deriv log(2|sin(θ)|) -log(2|sin(θ)|) Match?")
print("-" * 65)
for theta in angles:
series = lob_deriv_series(theta, 200)
log_formula = np.log(2 * np.abs(np.sin(theta)))
neg_log_formula = -np.log(2 * np.abs(np.sin(theta)))
match_pos = "YES" if abs(series - log_formula) < 0.01 else "NO"
match_neg = "YES" if abs(series - neg_log_formula) < 0.01 else "NO"
print(f"{theta:.1f} {series:+.6f} {log_formula:+.6f} {neg_log_formula:+.6f} {match_neg}")
print("\nConclusion: The correct formula is dΛ/dθ = -log(2|sin(θ)|)")
# Double check with finite differences
print("\n\nDouble-checking with finite differences:")
eps = 1e-7
for theta in [0.5, 1.0, 1.5]:
lob_plus = lob_series(theta + eps, 200)
lob_minus = lob_series(theta - eps, 200)
fd = (lob_plus - lob_minus) / (2 * eps)
series = lob_deriv_series(theta, 200)
print(f"θ={theta}: FD={fd:+.6f}, Series={series:+.6f}, Difference={abs(fd-series):.2e}") |