ψ ⁡ ( z + 1 2 ) − ψ ⁡ ( z 2 ) = 2 ⁢ ∑ k = 0 ∞ ( − 1 ) k k + z \psi \left({\frac {z+1}{2}}\right)-\psi \left({\frac {z}{2}}\right)=2\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k+z}}