Journal of Inequalities in Pure and
Applied Mathematics
THE ANALYTIC DOMAIN IN THE IMPLICIT FUNCTION THEOREM
H.C. CHANG, W. HE AND N. PRABHU
School of Industrial Engineering
Purdue University
West Lafayette, IN 47907
EMail: prabhu@ecn.purdue.edu
volume 4, issue 1, article 12,
2003.
Received 30 May, 2002;
accepted 12 September, 2002.
Communicated by: H.M. Srivastava
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
The Implicit Function Theorem asserts that there exists a ball of nonzero radius within which one can express a certain subset of variables, in a system
of analytic equations, as analytic functions of the remaining variables. We derive a nontrivial lower bound on the radius of such a ball. To the best of our
knowledge, our result is the first bound on the domain of validity of the Implicit
Function Theorem.
The Analytic Domain in the
Implicit Function Theorem
2000 Mathematics Subject Classification: 30E10
Key words: Implicit Function Theorem, Analytic Functions.
H.C. Chang, W. He and N. Prabhu
The first and third authors were supported in part by ONR grant N00014-96-1-0281
and NSF grant 9800053CCR. The second author was supported in part by ONR
grant N00014-96-1-0281.
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The authors also would like to acknowledge the help they received from Professors
Lempert and Catlin in the proof of Theorem 1.1.
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The Size of the Analytic Domain . . . . . . . . . . . . . . . . . . . . . . .
References
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1.
The Size of the Analytic Domain
The Implicit Function Theorem is one of the fundamental theorems in multivariable analysis [1, 4, 5, 6, 7]. It asserts that if ϕi (x, z) = 0, i = 1, . . . , m,
x ∈ Cn , z ∈ Cmare complex
analytic functions in a neighborhood of a point
ϕ1 ,...,ϕm (0) (0)
(x , z ) and J z1 ,...,zm (0) (0) 6= 0, where J is the Jacobian determinant,
(x
,z
)
then there exists an > 0 and analytic functions g1 (x), . . . , gm (x) defined in
the domain D = {x | kx − x(0) k < } such that ϕi (x, g1 (x), . . . , gm (x)) =
0, for i = 1, . . . , m in D. Besides its central role in analysis the theorem also
plays an important role in multi-dimensional nonlinear optimization algorithms
[2, 3, 8, 9]. Surprisingly, despite its overarching importance and widespread
use, a nontrivial lower bound on the size of the domain D has not been reported
in the literature and in this note, we present the first lower bound on the size of
D. The bound is derived in two steps. First we use Roche’s Theorem to derive
a lower bound for the case of one dependent variable – i.e., m = 1 – and then
extend the result to the general case.
Theorem 1.1. Let ϕ(x, z) be an analytic function of n + 1 complex variables,
x ∈ Cn , z ∈ C at (0, 0). Let ∂ϕ(0,0)
= a 6= 0, and let |ϕ(0, z)| ≤ M on B
∂z
where B = {(x, z)| k(x, z)k ≤ R}. Then z = g(x) is an analytic function of x
in the ball
1
M r2
(1.1) kxk ≤ Θ1 (M, a, R; ϕ) :=
|a| r − 2
,
M
R − rR
R |a| R2
where r = min
,
.
2 2M
The Analytic Domain in the
Implicit Function Theorem
H.C. Chang, W. He and N. Prabhu
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Proof. Since ϕ(x, z) is an analytic function of complex variables, by the Implicit Function Theorem z = g(x) is an analytic function in a neighborhood U
of (0, 0). To find the domain of analyticity of g we first find a number r > 0
such that ϕ(0, z) has (0,0) as its unique zero in the disc {(0, z) : |z| ≤ r}. Then
we will find a number s > 0 so that ϕ(x, z) has a unique zero (x, g(x)) in the
disc {(x, z) : |z| ≤ r} for |x| ≤ s with the help of Roche’s theorem. Then we
show that in the domain {x : kxk ≤ s} the implicit function z = g(x) is well
defined and analytic.
Note that if we expand the Taylor series of the function ϕ with respect to the
variable z, we get
∞
X
∂ϕ(0, 0)
ϕ(0, z) =
z+
∂z
j=2
∂ j ϕ(0,0) j
z
∂z j
j!
H.C. Chang, W. He and N. Prabhu
.
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Let us assume that | ∂ϕ(0,0)
| = a > 0. |ϕ(0, z)| ≤ M on B where B = {(x, z) :
∂z
k(x, z)k ≤ R} . Then by Cauchy’s estimate, we have
j
∂ ϕ(0,0) z j |z|j
∂zj
≤ j M.
j!
R
This implies that
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|ϕ(0, z)| ≥ |a| · |z| −
∞
X
j=2
(1.2)
The Analytic Domain in the
Implicit Function Theorem
= |a| · |z| −
M
|z|
R
M |z|2
.
R2 − |z|R
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2
|z|
Since our goal is to have |ϕ(0, z)| > 0, it is sufficient to have |a|·|z|− RM
2 −|z|R >
0. Dividing both sides by |z| we get
|a| >
M |z|
⇐⇒ |a|(R2 − |z|R) − M |z| > 0 ⇐⇒ |z|(|a|R + M ) < |a|R2
− |z|R
|a|R2
R
⇐⇒ |z| <
=
.
M
|a|R + M
1 + |a|R
R2
We next choose
r = min
= min
R
,
1+1
R
The Analytic Domain in the
Implicit Function Theorem
M
M
+ |a|R
|a|R
|a|R2
R
, 2M
2
H.C. Chang, W. He and N. Prabhu
.
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To compute s we need Roche’s Theorem.
Theorem 1.2 (Roche’s Theorem). [1] Let h1 and h2 be analytic on the open
set U ⊂ C, with neither h1 nor h2 identically 0 on any component of U . Let
γ be a closed path in U such that the winding number n(γ, z) = 0, ∀z ∈
/ U.
Suppose that
|h1 (z) − h2 (z)| < |h2 (z)|,
∀z ∈ γ.
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Then n(h1 ◦ γ, 0) = n(h1 ◦ γ, 0). Thus h1 and h2 have the same number of zeros
inside γ, counting multiplicity and index.
Close
Let h1 (z) := ϕ(0, z), and h2 := ϕ(x, z). We can treat x as a parameter, so
our goal is to find s > 0 such that if |x| < s, then
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J. Ineq. Pure and Appl. Math. 4(1) Art. 12, 2003
|ϕ(0, z) − ϕ(x, z)| < |ϕ(0, z)|,
∀z ∈ γ,
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where γ = {z : |z| = r}. We know |ϕ(0, z) − ϕ(x, z)| < M s if γ ⊂ B and we
|z|2
also have |ϕ(0, z)| > |a| · |z| − RM
2 −|z|R from (1.2). It is sufficient to have
M s < |a| · |z| −
M |z|2
.
R2 − |z|R
On γ, we know |z| = r, and therefore as long as
M r2
1
s<
|a|r − 2
,
M
R − rR
we can apply the Roche’s theorem and guarantee that the function ϕ(x, z) has
a unique zero, call it g(x). That is, ϕ(x, g(x)) = 0 and g(x) is hence a well
defined function of x.
Note that in Roche’s theorem, the number of zeros includes the multiplicity
and index. Therefore all the zeros we get are simple zeros since (0, 0) is a
simple zero for ϕ(0, z). This is because ϕ(0, 0) = 0 and ϕz (0, 0) 6= 0. Hence
we can conclude that for any x such that |x| < s, we can find a unique g(x) so
that ϕ(x, g(x)) = 0 and ϕz (x, g(x)) 6= 0.
We use Theorem 1.1 to derive a lower bound for m ≥ 1 below. Let ϕi (x, z) =
0, i = 1, . . . , m, x ∈ Cn , z ∈ Cm be analytic functions at (x(0) , z (0) ). Let
∂ϕ1 (x(0) ,z(0) )
∂ϕ1 (x(0) ,z (0) ) ···
∂z1
∂zm
.
.
(0) (0)
= am 6= 0
..
..
(1.3) Jm (x , z ) := ∂ϕm (x(0) ,z(0) )
∂ϕm (x(0) ,z (0) ) ···
∂z
∂z
1
m
The Analytic Domain in the
Implicit Function Theorem
H.C. Chang, W. He and N. Prabhu
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and let
(1.4)
ϕi (x(0) , z1 , . . . , zm ) ≤ M, for i = 1, . . . , m
on
(1.5)
B = {(x, z1 , . . . , zm ) | k(x, z) − (x(0) , z (0) )k ≤ R}.
Since Jm (x(0) , z (0) ) 6= 0, some (m − 1) × (m − 1) sub-determinant in the matrix
corresponding to Jm (x(0) , z (0) ) must be nonzero. Without loss of generality, we
may assume that
∂ϕ2 (x(0) ,z(0) )
∂ϕ2 (x(0) ,z (0) ) ·
·
·
∂z2
∂zm
.
.
(0) (0)
..
..
Jm−1 (x , z ) := (1.6)
(0)
(0)
∂ϕm (x(0) ,z(0) )
∂ϕm (x ,z ) ···
∂z2
∂zm
H.C. Chang, W. He and N. Prabhu
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= am−1 6= 0.
By induction we conclude that there exist analytic functions ψ2 (x, z1 ), . . . ,
(0)
ψm (x, z1 ) and that we can compute a Θm−1 (x(0) , z1 ; ϕ2 , . . . , ϕm ) > 0 such
that
ϕi (x, z1 , ψ2 (x, z1 ), . . . , ψm (x, z1 )) = 0, i = 2, . . . , m
in
(0)
The Analytic Domain in the
Implicit Function Theorem
(0)
Dn+1 := {(x, z1 ) | k(x, z1 ) − (x(0) , z1 )k ≤ Θm−1 (x(0) , z1 ; ϕ2 , . . . , ϕm )}.
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Define
J. Ineq. Pure and Appl. Math. 4(1) Art. 12, 2003
(1.7)
Γ(x, z1 ) := ϕ1 (x, z1 , ψ2 (x, z1 ), . . . , ψm (x, z1 )).
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Then we have
m
∂Γ
∂ϕ1 X ∂ϕ1 ∂ψi
=
+
·
.
∂z1
∂z1
∂z
∂z
i
1
i=2
(1.8)
Since ϕ2 (x, z1 , ψ2 , . . . , ψm ) = 0, . . . , ϕm (x, z1 , ψ2 , . . . , ψm ) = 0 in Dn+1 , differentiating with respect to z1 we have
m
∂ϕi X ∂ϕi ∂ψj
∂ϕi
=
+
·
= 0; i = 2, . . . , m
∂z1
∂z1 j=2 ∂zj ∂z1
The Analytic Domain in the
Implicit Function Theorem
H.C. Chang, W. He and N. Prabhu
or in other words
 ∂ϕ2
∂z2
(1.9)


···
..
.
∂ϕm
∂z2
∂ϕ2
∂zm
..
.
···
∂ϕm
∂zm



∂ψ2
∂z1
..
.
∂ψm
∂z1
Using Cramer’s rule and (1.9) we have
∂ϕ2
∂ϕ2
∂ϕ2
∂z2 · · · ∂zi−1 ∂z1
..
..
..
.
.
.
∂ϕm
∂ϕm
∂ϕm
· · · ∂zi−1 ∂z1
∂ψi
∂z2
(1.10)
=−
∂z1
Jm−1




 = −
∂ϕ2
∂zi+1
···
..
.
∂ϕm
∂zi+1
∂ϕ2
∂z1
..
.
∂ϕm
∂z1

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
.
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∂ϕ2
∂zm
..
.
···
∂ϕm
∂zm
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; i = 2, . . . , m.
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Substituting (1.10) into (1.8) and simplifying we get
(0) (0)
∂ϕ1 (x(0) ,z(0) )
· · · ∂ϕ1 (x∂zm,z )
∂z
1
..
..
.
.
(0) ,z (0) )
(0) ,z (0) )
∂ϕ
(x
∂ϕ
(x
m
m
(0)
···
∂Γ(x(0) , z1 )
∂z1
∂zm
=
(0)
(0)
∂z1
Jm−1 (x , z )
(0) (0)
am
Jm (x , z )
=
=
6= 0.
Jm−1 (x(0) , z (0) )
am−1
Therefore we can apply Theorem 1.1 to Γ(x, z1 ) and conclude that there exists
an implicit function z1 = g1 (x) in
Dn := x ∈ Cn kx − x(0) k
am
(0) (0)
, min R, Θm−1 (x , z1 ; ϕ2 , . . . , ϕm ) ; ϕ1
≤ Θ1 M,
am−1
such that in Dn , ϕi (x, g1 (x), g2 (x), . . . , gm (x)) = 0, i = 1, . . . , m where
gj (x) := ψj (x, g1 (x)), j = 2, . . . , m.
In summary, the sought lower bound on the size of the analytic domain of
implicit functions is expressed recursively as
(1.11) Θm (x(0) , z (0) ; ϕ1 , . . . , ϕm )
am
(0) (0)
= Θ1 M,
, min(R, Θm−1 (x , z1 ; ϕ2 , . . . , ϕm )); ϕ1
am−1
The Analytic Domain in the
Implicit Function Theorem
H.C. Chang, W. He and N. Prabhu
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using the definition of Θ1 in Theorem 1.1 and of M, am , am−1 and R in equations (1.4), (1.3), (1.6) and (1.5) respectively.
The Analytic Domain in the
Implicit Function Theorem
H.C. Chang, W. He and N. Prabhu
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References
[1] R.B. ASH, Complex Variables, Academic Press, 1971.
[2] D.P. BERTSEKAS, Nonlinear Programming, Athena Scientific Press,
1999.
[3] R. FLETCHER, Practical Methods of Optimization, John Wiley and
Sons, 2000.
[4] R.C. GUNNING, Introduction to Holomorphic Functions of Several
Variables: Function Theory, CRC Press, 1990.
The Analytic Domain in the
Implicit Function Theorem
H.C. Chang, W. He and N. Prabhu
[5] L. HORMANDER, Introduction to Complex Analysis in Several Variables, Elsevier Science Ltd., 1973.
[6] S.G. KRANTZ, Function Theory of Several Complex Variables, WileyInterscience, 1982.
[7] R. NARASIMHAN, Several Complex Variables, University of Chicago
Press, 1974.
[8] S. NASH AND A. SOFER, Linear and Nonlinear Programming,
McGraw-Hill, 1995.
[9] J. NOCEDAL
Verlag, 1999.
AND
S.J. WRIGHT, Numerical Optimization, Springer
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