Would the complex function sech(z) be holomorphic because cosh(z) defined by its power series is holomorphic? I am not sure why sech (z) is even (complex) differentiable everywhere since surely it is not a "flat map" and the "gradient" along a contour line ($=0$) and that along a perpendicular of a contour line ($\neq 0$) are different! Please explain!
Also is there a effective way of telling whether a complex function is differentiable and/or holomorphic?