This document provides an overview of Liouville's theorem and Gauss's mean value theorem from complex analysis. It includes the statements of both theorems, outlines their proofs, and provides examples and frequently asked questions about Liouville's theorem. Specifically, Liouville's theorem states that every bounded entire function must be constant, while Gauss's mean value theorem relates the derivative of a function to the average value of the derivative over an interval.

1. Liouville's theorem and gauss’s
mean value theorem
Presented By: Supervisor:
Dastan Ahmed
Darstan Hasan
Faculty of Science
Department of Mathematics

2. Outline
•Introduction
•Proof of Liouville’s Theorem
•Corollaries of Liouville’s Theorem
•Related Articles
•Solved Examples on Liouville’s Theorem
•Frequently Asked Questions on Liouville’s Theorem
•What is Liouville’s Theorem in complex analysis?
•What are the requirements of Liouville’s theorem?
•What is the statement of Liouville’s theorem?
•Liouville’s theorem is named after which mathematician?
•Gauss Law Formula
•References

3. Introduction
In complex analysis, Liouville's theorem, named after Joseph Liouville (although
the theorem was first proven by Cauchy in 1844), states that
every bounded entire function must be constant. That is, every holomorphic
function for which there exists a positive number such that for all in is
constant. Equivalently, non-constant holomorphic functions on have unbounded
images. The theorem is considerably improved by Picard's little theorem, which
says that every entire function whose image omits two or more complex
numbers must be constant.

4. According to Liouville’s Theorem, if f is an integral function (entire function)
satisfying the inequality |f(z)| ≤ M, where M is a positive constant, for all values of z
in complex plane C, then f is a constant function Liouville’s theorem is concerned
with the entire function being bounded over a given domain in a complex plane. An
entire or integral function is a complex analytic function that is analytic throughout
the whole complex plane. For example, exponential function, sin z, cos z and
polynomial functions. The statement of Liouville’s Theorem has several versions.

5. Thus according to Liouville’s Theorem, if f is an integral function (entire function)
satisfying the inequality |f(z)| ≤ M, where M is a positive constant, for all values of z
in complex plane C, then f is a constant function.In other words, if f(z) is an analytic
function for all finite values of z and is bounded for all values of z in C, then f is a
constant function.In mathematics, the mean value theorem (or Lagrange theorem)
states, roughly, that for a given planar arc between two endpoints, there is at least
one point at which the tangent to the arc is parallel to the secant through its
endpoints. It is one of the most important results in real analysis.

6. This theorem is used to prove statements about a function on
an interval starting from local hypotheses about derivatives at points of the
interval. More precisely, the theorem states that if is a continuous function on
the closed interval and differentiable on the open interval, then there exists a
point in such that the tangent at is parallel to the secant line through the
endpoints . Gauss Law states that the total electric flux out of a closed surface
is equal to the charge enclosed divided by the permittivity. The electric flux in
an area is defined as the electric field multiplied by the area of the surface
projected in a plane and perpendicular to the field.

7. Proof of Liouville’s Theorem
By the theorem hypothesis, f is bounded entire function such that for M be a
positive constant |f(z)| ≤ M.Let z1 and z2 be arbitrary points in z-plane. C be a
circle in z-plane with z1 as centre and radius R such that z2 be any point inside
the circle C. Then by Cauchy’s Integral Formula, we have

8. We can choose R large enough as f is analytic throughout z-plane, so that |z2 – z1|
< R/2.
Since the circle C:|z – z1| < R, we have
|z – z2| = |(z – z1) – (z2 – z1)| ≥ |z – z1| – |z2 – z1| ≥ R – R/2 = R/2
⇒ |z – z2| ≥ R/2
Now, from (1), we have

9. The right-hand side of the above inequality tends to zero as R → ∞ . Hence, for the
entire function f, R→ ∞ , therefore
f(z2) – f(z1) = 0 ⇒ f(z1) = f(z2).
Since z1 and z2, are arbitrarily chosen, this holds for every points in the complex z-
plane.Thus, f is a constant function.

10. Corollaries of Liouville’s Theorem
 A non constant entire function is not bounded.
 The fundamental theorem of Algebra: Every non constant complex polynomial
has a root.
 If f is a non constant entire function, then w-image if f is dense in complex
plane C.

11. Related Articles
 Complex Numbers
 Analytic Functions
 Limits and Continuity
 Differentiabilty
 Integration
 Complex Conjugate

12. Solved Examples on Liouville’s
Theorem
Example 1:
Let f = u(z) + iv(z) be an entire function in complex plane C. If |u(z)| < M for every
z in C, where M is a positive constant, then prove that f is a constant function.
Solution:
Given, f = u(z) + iv(z) is an entire function in complex plane C such that |u(z)| < M
for every z in C.
Let g(z) = ef(z)
since f is entire ⇒ g is also an entire function

13. Now, takin modulus on both side
|g(z)| = |ef(z)| = |eu(z) + iv(z)| = |eu(z) . eiv(z)| = |eu(z)|
Since, |ei𝜃| = 1
Therefore, |g(z)| = eM, (as |u(z)| < M, hence eM is constant)
⇒ g(z) is a bounded function
As g(z) is bounded entire function
By Liouville’s theorem, g is a constant function
⇒ ef(z) is a constant function
⇒ f = u(z) + iv(z) is constant
⇒f is a constant function.

14. Example 2:
Let f be an entire function such that |f(z)| ≥ 1 for every z in C.
Prove that f is a constant function.
Solution:
Given f is an entire function such that |f(z)| ≥ 1 for every z in C
Let g(z) = 1/f(z)

15. Since f is an entire function ⇒ g is an entire function
Now, |g(z)| = |1/f(z)| = 1/|f(z)|
As |f(z)| ≥ 1 ⇒ 1/|f(z)| ≤ 1
Therefore, |g(z)| ≤ 1
⇒ g is bounded
Thus, g is an bounded entire function
Then, by Liouville’s Theorem g is a constant function
Consequently, f is a constant function.

16. Frequently Asked Questions on
Liouville’s Theorem
What is Liouville’s Theorem in complex analysis?
According to Liouville’s theorem, a bounded entire function is a constant
function.
What are the requirements of Liouville’s theorem?
To satisfy the condition of Liouville’s theorem, the function has to be an entire
function as well as bounded for all values of z in z-plane.

17. What is the statement of Liouville’s theorem?
According to Liouville’s Theorem, if f is an integral function (entire function)
satisfying the inequality |f(z)| ≤ M, where M is a positive constant, for all values of z
in complex plane C, then f is a constant function.
Liouville’s theorem is named after which mathematician?
Liouville’s theorem is named after a French mathematician and Engineer Joseph
Liouville.

18. Gauss Law Formula
As per the Gauss theorem, the total charge enclosed in a closed surface is
proportional to the total flux enclosed by the surface. Therefore, if ϕ is total flux
and ϵ0 is electric constant, the total electric charge Q enclosed by the surface is;
Q = ϕ ϵ0

19. The Gauss law formula is expressed by;
ϕ = Q/ϵ0
Where,
Q = total charge within the given surface,
ε0 = the electric constant.
⇒ Also Read: Equipotential Surface

20. The Gauss Theorem
The net flux through a closed surface is directly proportional to the net charge in
the volume enclosed by the closed surface.
Φ = → E.d → A = qnet/ε0

21. In simple words, the Gauss theorem relates the ‘flow’ of electric field lines (flux)
to the charges within the enclosed surface. If no charges are enclosed by a
surface, then the net electric flux remains zero.
This means that the number of electric field lines entering the surface equals
the field lines leaving the surface.

22. The Gauss theorem statement also gives
an important corollary
The electric flux from any closed surface is only due to the sources (positive
charges) and sinks (negative charges) of the electric fields enclosed by the
surface. Any charges outside the surface do not contribute to the electric flux.
Also, only electric charges can act as sources or sinks of electric fields. Changing
magnetic fields, for example, cannot act as sources or sinks of electric fields.

23. Problems on Gauss Law
Problem 1: A uniform electric field of magnitude E = 100 N/C exists in the space
in the X-direction. Using the Gauss theorem calculate the flux of this field
through a plane square area of edge 10 cm placed in the Y-Z plane. Take the
normal along the positive X-axis to be positive.

24. Solution:
The flux Φ = ∫ E.cosθ ds.
As the normal to the area points along the electric field, θ = 0.
Also, E is uniform so, Φ = E.ΔS = (100 N/C) (0.10m)2 = 1 N-m2.

25. Problem 2:
A large plane charge sheet having surface charge density σ = 2.0 × 10-6 C-m-2 lies
in the X-Y plane. Find the flux of the electric field through a circular area of radius
1 cm lying completely in the region where x, y, and z are all positive and with its
normal, making an angle of 600 with the Z-axis.

26. Solution:
The electric field near the plane charge sheet is E = σ/2ε0 in the direction away from
the sheet. At the given area, the field is along the Z-axis.
The area = πr2 = 3.14 × 1 cm2 = 3.14 × 10-4 m2.
The angle between the normal to the area and the field is 600.
Hence, according to Gauss theorem, the flux
= E.ΔS cos θ
= σ/2ε0 × pr2 cos 60º
= 17.5 N-m2C-1.

27. Problem 3:
A charge of 4×10-8 C is distributed uniformly on the surface of a sphere of radius 1
cm. It is covered by a concentric, hollow conducting sphere of radius 5 cm.
Find the electric field at a point 2 cm away from the centre.
A charge of 6 × 10-8C is placed on the hollow sphere. Find the surface charge density
on the outer surface of the hollow sphere.

28. References
1. Solomentsev, E.D.; Stepanov, S.A.; Kvasnikov, I.A. (2001) [1994], "Liouville
theorems", Encyclopedia of Mathematics, EMS Press
2. Nelson, Edward (1961). "A proof of Liouville's theorem". Proceedings of the
AMS. 12 (6): 995. doi:10.1090/S0002-9939-1961-0259149-4.
3. Benjamin Fine; Gerhard Rosenberger (1997). The Fundamental Theorem of
Algebra. Springer Science & Business Media. pp. 70–71. ISBN 978-0-387-
94657-3.

29. 4. Liouville, Joseph (1847), "Leçons sur les fonctions doublement périodiques",
Journal für die Reine und Angewandte Mathematik (published 1879), vol. 88, pp.
277–310, ISSN 0075-4102, archived from the original on 2012-07-11
5. Cauchy, Augustin-Louis (1844), "Mémoires sur les fonctions complémentaires",
Œuvres complètes d'Augustin Cauchy, 1, vol. 8, Paris: Gauthiers-Villars (published
1882)
6. Lützen, Jesper (1990), Joseph Liouville 1809–1882: Master of Pure and Applied
Mathematics, Studies in the History of Mathematics and Physical Sciences, vol.
15, Springer-Verlag, ISBN 3-540-97180-7
7. a concise course in complex analysis and Riemann surfaces, Wilhelm Schlag,
corollary 4.8, p.77 http://www.math.uchicago.edu/~schlag/bookweb.pdf
Archived 2017-08-30 at the Wayback Machine
8. Denhartigh, Kyle; Flim, Rachel (15 January 2017). "Liouville theorems in the Dual
and Double Planes". Rose-Hulman Undergraduate Mathematics Journal. 12 (2).