Which of the following functions are entire? Give reasons.
(i) f(z) = sin |z|,
(ii) f(z) = $\displaystyle z^5 + iz^7 + z^{12},$
(iii) f(z) = f(x + iy) = x,
(iv) f(z) = 2.
This function is bounded, so if it was entire then it would be a constant.
This is a contradiction so it is not entire.
Polynomials are always entire.(ii) f(z) = $\displaystyle z^5 + iz^7 + z^{12},$
Check the Cauchy-Riemann equations(iii) f(z) = f(x + iy) = x
Polynomials are always entire.(iv) f(z) = 2.