Research proof index

An adjointness identity for difference convolution. The inner product of $f$ with $g\mathbin{\circleddash}h$ equals the inner product of $\overline g$ with $\overline f*\overline h$.

Monotonicity of weighted L^p norms in the exponent, for probability weights.

The discriminant Δ_fun = 1728⁻¹(E₄³ - E₆²) equals the standard discriminant Δ.

Θ₂(It) has positive real part for t > 0. Proof: Each term Θ₂_term n (It) = exp(-π(n+1/2)²t) is a positive real. The sum of positive reals is positive.

Each term Θ₂_term n (I*t) has positive real part equal to exp(-π(n+1/2)²t) for t > 0.

Each term Θ₄_term n (I*t) has zero imaginary part for t > 0.