Final answer:
G(z;u) is an analytic function for all z ≠ 0 and has an essential singularity at z = 0 if and only if u ≠ 0.
Step-by-step explanation:
Understanding the Analytic Nature and Singularity of a Complex Function
To demonstrate that G(z;u) = exp[(2u)(z - z1)], for z ≠ 0, is an analytic function and has an essential singularity at z = 0, we look into the function's behavior and composition. Since the exponential function is known to be analytic everywhere, the function G(z;u) will also be analytic wherever it is well-defined, which is for all z ≠ 0. The absence of poles in the function contributes to this property. The question further asks to show that there is an essential singularity at z = 0 unless u = 0. This follows from the nature of the exponential function, which, when multiplied by a nonzero scalar and expanded about z = 0, would lead to terms of arbitrarily negative powers of z, indicating the presence of an essential singularity as per the definition in complex analysis.
In conclusion, G(z;u) is indeed an analytic function for all z ≠ 0 and possesses an essential singularity at z = 0 if and only if u ≠ 0.