§ 1. Integrals of Holomorphic Functions
5
We often write
df (c; h) = f
(c)h
instead of f
(c)h; for h = (u, v), df (c; h) = u clearly holds if f is the coordinate function (x, y) → x, df (c; h) = v if f is (x, y) → y and df (c; h) = h
if f is
4 the identity map z = (x, y) → (x, y). We can, therefore, write
df [c; dz(x; h)] = D 1 f (c)dx(c; h) + D 2 f (c)dy(c; h)
or
df (c; dz) = D 1 f (c)dx + D 2 f (c)dy
(1.5)
for short.
We will also need the chain rule in two cases.
(a) Suppose that f is of class C
1 on U and let μ : I −→ U be a function
defined on an interval I of R, whence a composite function p = f ◦ μ : t →
f [μ(t)]. The same holds for the function p at each point t at which μ is
differentiable and
p
(t) = f
[μ(t)] μ
(t)
(1.6)
is the image of the vector
5 μ
(t) ∈ R
2 under the linear map tangent to f at
μ(t). If f and μ are of class C
1 , so is p; if f is of class C
1 and μ of class C
1/2 ,
the obviously continuous function p is of class C
1/2 since p
(t) exists outside
some countable set and as the product of the continuous function f
[μ(t)]
and the regulated function μ
(t) is regulated. The function p is, therefore, a
primitive for f
[μ(t)] μ
(t).
(b) If g is a map from an open set V ⊂ R
2 to U , whence again a composite
map p = f ◦ g : V −→ R
2 , then, at each point c ∈ V where g is differentiable,
so is p, and
p
(c) = f
[g(c)] ◦ g
(c)
(1.7)
is the composite or product of linear maps tangent to g at c and to f at g(c).
4 Here the letter z represents the point with x, y coordinates in R
2 rather than
the complex number x + iy. It is in the theory of holomorphic functions that it
is essential to regard points in the plane as complex numbers. Having said that,
using the letter z to represent a point in R
2 or in any other set is not forbidden.
5 If μ(t) = (μ1(t), μ2(t)), μ
(t) is the vector (μ
1 (t), μ
2 (t)). If we do not distinguish
between a point or a vector (u, v) ∈ R
2 and the complex number u + iv ∈ C,
the complex number μ
(t) becomes the usual derivative of the complex valued
function μ(t). However, this interpretation is not generally compatible with formula (6), since in the latter f
[μ(t)] is a linear map from R
2 to R
2 and not a
mere complex number. It is only if f is holomorphic that that three derivatives
occurring in (6) can be interpreted as complex numbers. The possibility of interpreting elements of R
2 in these two different ways often leads to confusion that,
for good reason, does not occur in R
n , when n ≥ 3.
5
We often write
df (c; h) = f
(c)h
instead of f
(c)h; for h = (u, v), df (c; h) = u clearly holds if f is the coordinate function (x, y) → x, df (c; h) = v if f is (x, y) → y and df (c; h) = h
if f is
4 the identity map z = (x, y) → (x, y). We can, therefore, write
df [c; dz(x; h)] = D 1 f (c)dx(c; h) + D 2 f (c)dy(c; h)
or
df (c; dz) = D 1 f (c)dx + D 2 f (c)dy
(1.5)
for short.
We will also need the chain rule in two cases.
(a) Suppose that f is of class C
1 on U and let μ : I −→ U be a function
defined on an interval I of R, whence a composite function p = f ◦ μ : t →
f [μ(t)]. The same holds for the function p at each point t at which μ is
differentiable and
p
(t) = f
[μ(t)] μ
(t)
(1.6)
is the image of the vector
5 μ
(t) ∈ R
2 under the linear map tangent to f at
μ(t). If f and μ are of class C
1 , so is p; if f is of class C
1 and μ of class C
1/2 ,
the obviously continuous function p is of class C
1/2 since p
(t) exists outside
some countable set and as the product of the continuous function f
[μ(t)]
and the regulated function μ
(t) is regulated. The function p is, therefore, a
primitive for f
[μ(t)] μ
(t).
(b) If g is a map from an open set V ⊂ R
2 to U , whence again a composite
map p = f ◦ g : V −→ R
2 , then, at each point c ∈ V where g is differentiable,
so is p, and
p
(c) = f
[g(c)] ◦ g
(c)
(1.7)
is the composite or product of linear maps tangent to g at c and to f at g(c).
4 Here the letter z represents the point with x, y coordinates in R
2 rather than
the complex number x + iy. It is in the theory of holomorphic functions that it
is essential to regard points in the plane as complex numbers. Having said that,
using the letter z to represent a point in R
2 or in any other set is not forbidden.
5 If μ(t) = (μ1(t), μ2(t)), μ
(t) is the vector (μ
1 (t), μ
2 (t)). If we do not distinguish
between a point or a vector (u, v) ∈ R
2 and the complex number u + iv ∈ C,
the complex number μ
(t) becomes the usual derivative of the complex valued
function μ(t). However, this interpretation is not generally compatible with formula (6), since in the latter f
[μ(t)] is a linear map from R
2 to R
2 and not a
mere complex number. It is only if f is holomorphic that that three derivatives
occurring in (6) can be interpreted as complex numbers. The possibility of interpreting elements of R
2 in these two different ways often leads to confusion that,
for good reason, does not occur in R
n , when n ≥ 3.