Complex Dierentiability, Holomorphy, Analyticity

Because complex functions of a complex variable represent mappings from R² to itself, it is best to frame the notion of the derivative in terms of multivariable derivatives. Some important denitions and theorems from calculus of several variables are now stated.

Denition (dierentiable at a point in the plane). f : R² → R² is dierentiable at some point a ∈ R² if and only if a matrix Df(a) exists and

x→alim

kR(x)k kx − ak

where R(x) = f(x)−(f(a) + Df(a) (x − a)). The matrix Df(a) is called the real derivative matrix or the real total derivative of f at a.

Denition (dierentiable on a region of the plane). For some open set U ⊆ R², f : U → R² is dierentiable on U if and only if f is dierentiable at every point of U.

It can be shown that dierentiability entails that the structure of Df(a) obtains a particular form. So that, if

f (x, y) = u(x, y) v(x, y)

 is dierentiable at a, then

Df (a) =

Denition (continuous of order n). For an open set U ⊆ R², f : U → R² is continuous of order n on U, written

f ∈ Cⁿ(U )

if and only if all partial derivatives of f of order up to and including n exist and are continuous. In particular, f ∈ C(U), i.e., f is continuous of order zero, if and only if f is continuous on U; f ∈ C¹(U ) if and only if all partials ∂fi/∂x_j, for i, j ∈ {1, 2}, exist and are continuous on U.

It can be shown that f ∈ C¹ entails that f is dierentiable. It can also be shown that dierentiability implies continuity, i.e. that f ∈ C, but that the derivative exists but is not necessarily continuous. Finally, we simply write down the inverse function theorem without proof,

Theorem A.3. For a function f : U ⊆ R² → R² with f ∈ C¹ and Df(a) invertible at a, the domain of f may be restricted to some open set V ⊆ U with a ∈ V so that f is invertible, f⁻¹ ∈ C¹, and Df⁻¹(f (a)) = Df (a)⁻¹

With these concepts in hand, it is fruitful to interpret complex function of a complex variable as derivatives of real functions from the plane to the plane. This is the source of dierent expressions used to discuss the dierentiability of such functions. These notions turn out to be equivalent via several theorems, nevertheless, the distinctions should be preserved so that the appropriate language is employed when one aspect over another is to be emphasized in any given situation.

A function f : U ⊆ C → C, is dierentiable, or, more specically, real dierentiable, whenever, taken as a function from R² to R², it is real dierentiable as dened above.

In this case, the matrix Df(a) exists and is dened as discussed above. Several partial derivatives are now proposed in a natural way and employed in the discussion of complex dierentiability and holomorphy.

Thus, these partials are simply corresponding columns of the derivative matrix Df(a).

Now, if x = (z + z)/2 and y = (z − z)/2i, take partial derivatives of f with respect to z and z via chain rule. Obtain the following

∂f

so and, for derivative with respect to the conjugate of z,

∂f

which motivates the following denition for partial derivatives of f with respect to z and with respect to z,

Multiples of equations (A.31), (A.32), (A.33), and (A.34) may be added and subtracted to obtain

Also, from these denitions, the following natural formulae also hold, ^∂z_∂z = 1, ^∂z_∂z = 1,

∂z

∂z = 0, ^∂z_∂z = 0, and, for any constant a ∈ C, ^∂c_∂z = ^∂a_∂z = 0.

Denition (complex dierentiable). f : U ⊆ C → C is complex dierentiable at a ∈ C if and only if the limit

df

exists and we say that the limit, ^df_dz

a, sometimes written f⁰(a), is the derivative of f at a.

Denition (holomorphic). f : U ⊆ C → C is holomorphic on U if and only if f is complex dierentiable at every point of U.

Sometimes, instead of complex dierentiable, some authors will say that the function is holomorphic at a point. Here, holomorphy refers to complex dierentiability on an open set which is the domain of the function. Some authors use analytic instead of these, but, f is analytic at a point z ∈ C if and only if the Taylor series of f converges to f on a neighborhood of z. It turns out that wherever f is holomorphic, derivatives of every order of f exist, and consequently that f is analytic there. So, via a theorem, these notions are equivalent, however, they are employed when dierent aspects of dierentiability are to be emphasized.

Denition (anti-holomorphic). For f : U ⊆ C → C, when the limit df

dz   a

= lim

z→a

f (z) − f (a)

z − a (A.40)

exists for every point a ∈ U, and we say that f is anti-holomorphic on U.

Theorem A.4. For an open set U ⊆ C and function f : U → C, the following are equivalent:

1. f is holomorphic on U,

2. f is dierentiable (in the real sense) but ^∂f_∂z = 0, and 3. ^∂u_∂x = ^∂v_∂y and ^∂u_∂y = −^∂v_∂x

consequently, if any one of the above hold, we may write df

dz = ∂f

∂z by the denitions of partials above.

Theorem A.5. For an open set U ⊆ C and function f : U → C, the following are equivalent:

1. f is anti-holomorphic on U,

2. f is dierentiable (in the real sense) but ^∂f_∂z = 0, and 3. ^∂u_∂x = −^∂v_∂y and ^∂u_∂y = ^∂v_∂x

consequently, if any ones of these hold, we may write df

dz = ∂f

∂z by the denitions of partials above.

The equations in (3) in theorem A.4 are known as the Cauchy-Riemann Equations and turn out to be extremely important in the subsequent discussion of harmonic functions, they are repeated here in rectangular form for convenience

∂u

∂x = ∂v

∂y and ∂v

∂x = −∂u

∂y (A.41)

Finally, the complex form of the inverse function theorem is stated here without proof, the proof follows from the result for real functions on the plane.

Theorem A.6 (Inverse Function Theorem). f : U ⊆ C → C holomorphic in U and f⁰(a) 6= 0 for some a ∈ U entails that f is invertible on a restriction of its domain, and the inverse f⁻¹ will be holomorphic near f(a) with

d dzf⁻¹

    f (z)

= 1

f⁰(z) on a neighborhood of f(a).