= Solution
The exponential definitions of <hyperbolic sine> and <hyperbolic cosine>, together with <Euler's formula>, give $\sinh(iu)=i\sin u$ and $\cosh(iu)=\cos u$. The argument of the <complex logarithm> therefore reduces to $i\sin(\pi/2)+\cos(9\pi/2)=i$.
Using the <principal logarithm>, $\log i=i\pi/2$, so
$$
\boxed{\Re z=0,\qquad\Im z=\frac\pi2\quad\text{(principal value)}.}
$$
If no branch of the <complex logarithm> is selected, its complete set of values is $i(\pi/2+2\pi k)$, $k\in\mathbb Z$. The <real part> remains zero, while the <imaginary part> is $\pi/2+2\pi k$. The original paper does not explicitly state a branch convention.
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