the permanence of functional relations
If an algebraic identity holds for real numbers, does it still hold once you replace the real variable by a complex one? The principle of permanence of functional relations says yes, automatically, provided the functions involved are analytic and the identity holds on a set rich enough to invoke the identity theorem. An equation true on the real line is true on the whole complex plane, for free, with no recomputation. It is the identity theorem dressed up as a working tool.
Here is the mechanism. Suppose you have an identity like F(z) = G(z), where F and G are built analytically from familiar functions, and you already know F = G for all real z (or on any set with an interior limit point). Form the difference F - G: it is analytic and vanishes on a set with a limit point, so by the identity theorem it vanishes identically, giving F = G for all complex z. The real-variable proof you already trust does double duty: it establishes the identity on the reals, and permanence carries it to the plane. The catch is that both sides must genuinely be analytic — if a relation involves the modulus, the conjugate, or a branch choice, it may not extend, because those operations are not analytic.
This principle quietly underwrites half of the elementary computations in the subject. The addition formula e^(z+w) = e^z e^w, the Pythagorean identity sin^2 z + cos^2 z = 1, the relation e^(iz) = cos z + i sin z, and de Moivre's formula all start as real or algebraic facts and become complex theorems by permanence. It saves enormous labour: instead of re-deriving every identity in the complex setting, you verify it where it is easy and let analyticity do the rest. But always check analyticity first — permanence is a privilege of analytic relations only.
We know e^(x+y) = e^x e^y for all real x, y. Fix y and view both sides as analytic functions of x; they agree on the real axis, so by permanence they agree for all complex z: e^(z+y) = e^z e^y. Now fix that complex z and vary y analytically — agreement on the real y-axis spreads to all complex w, giving e^(z+w) = e^z e^w everywhere.
A two-step permanence argument lifts a real identity to a fully complex one, one variable at a time.
Permanence applies only to relations between analytic functions. Identities involving |z|, the conjugate z-bar, or 'the' real part are NOT analytic and generally do not persist — for instance |e^z| = e^x, not e^(Re z) treated as analytic.