Series for log(z+a) for positive a and Re(z) greater than -a

Theorem

The following formula holds for $a > 0, \mathrm{Re}(z) \geq -a$, and $z \neq -a$: $$\log(z+a) = \log(a) + 2 \displaystyle\sum_{k=0}^{\infty} \left( \dfrac{z}{2a+z} \right)^{2k+1} \dfrac{1}{2k+1},$$ where $\log$ denotes the logarithm.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.29$