G₀–G₄ Complex Geometry

Shared analytic structure of the five G-functions: zeros, poles, critical points, lattice directions, periods, and shift symmetries.

Shared z-plane
z = 0 G₀: zero G₀: critical G₁: pole G₂: pole G₄: criticalz = 2πi(-1) G₀: criticalz = 2πi(1) G₀: criticalz = −7 arcosh(1π) − 7πi/2 G₁: polez = 7 arcosh(1π) − 7πi/2 G₁: polez = −7 arcosh(1π) + 7πi/2 G₁: polez = 7 arcosh(1π) + 7πi/2 G₁: polez = −4(1) G₂: polez = −4(2) G₂: polez = −4(3) G₂: polez = −4(4) G₂: polez = 5(-15)π/14 G₃: zero G₄: zero G₄: criticalz = 5(-13)π/14 G₃: zero G₄: zero G₄: criticalz = 5(-11)π/14 G₃: zero G₄: zero G₄: criticalz = 5(-9)π/14 G₃: zero G₄: zero G₄: criticalz = 5(-7)π/14 G₃: zero G₄: zero G₄: criticalz = 5(-5)π/14 G₃: zero G₄: zero G₄: criticalz = 5(-3)π/14 G₃: zero G₄: zero G₄: criticalz = 5(-1)π/14 G₃: zero G₄: zero G₄: criticalz = 5(1)π/14 G₃: zero G₄: zero G₄: criticalz = 5(3)π/14 G₃: zero G₄: zero G₄: criticalz = 5(5)π/14 G₃: zero G₄: zero G₄: criticalz = 5(7)π/14 G₃: zero G₄: zero G₄: criticalz = 5(9)π/14 G₃: zero G₄: zero G₄: criticalz = 5(11)π/14 G₃: zero G₄: zero G₄: criticalz = 5(13)π/14 G₃: zero G₄: zero G₄: criticalz = 5(15)π/14 G₃: zero G₄: zero G₄: criticalz = -17πi/5 G₃: zeroz = -15πi/5 G₃: zeroz = -13πi/5 G₃: zeroz = -11πi/5 G₃: zeroz = -9πi/5 G₃: zeroz = -7πi/5 G₃: zeroz = -5πi/5 G₃: zeroz = -3πi/5 G₃: zeroz = -1πi/5 G₃: zeroz = 1πi/5 G₃: zeroz = 3πi/5 G₃: zeroz = 5πi/5 G₃: zeroz = 7πi/5 G₃: zeroz = 9πi/5 G₃: zeroz = 11πi/5 G₃: zeroz = 13πi/5 G₃: zeroz = 15πi/5 G₃: zeroz = 17πi/5 G₃: zeroz = 5π(-14)/14 G₄: criticalz = 5π(-12)/14 G₄: criticalz = 5π(-10)/14 G₄: criticalz = 5π(-8)/14 G₄: criticalz = 5π(-6)/14 G₄: criticalz = 5π(-4)/14 G₄: criticalz = 5π(-2)/14 G₄: criticalz = 5π(2)/14 G₄: criticalz = 5π(4)/14 G₄: criticalz = 5π(6)/14 G₄: criticalz = 5π(8)/14 G₄: criticalz = 5π(10)/14 G₄: criticalz = 5π(12)/14 G₄: criticalz = 5π(14)/14 G₄: critical