  We discuss the validity of generalized Debye-H\"uckel (GDH) equation proposed
by Fisher {\itshape et al.} from the functional integral point of view. The GDH
theory considers fluctuations around prescribed densities of positive and
negative charges. Hence we first formulate a density functional integral
expression for the canonical system of Coulomb gas, and also demonstrate that
this is a dual form to the Sine-Gordon theory. Our formalism reveals the
following: (i) The induced charge distribution around supposed density favors
not only the cancellation of additional electrostatic potential like the
original DH theory, but also the countervailing of chemical potential
difference between imposed and equilibrium value. (ii) As a consequence
apparent charge, absent in the GDH equation, comes out in our generalized
equation. (iii) That is, the GDH equation holds only in special cases.
