CR-Equations
and
Harmonic
functions
Dr. Hina Dutt
hina.dutt@seecs.edu.pk
SEECS-NUST
Advanced
Engineering
Mathematics (10th
Edition) by Ervin
Kreyszig
A First Course in
Complex Analysis
with Applications by
Dennis G. Zill and
Patrick D. Shanahan.
• Chapter: 13
• Sections: 13.4
• Chapter: 3
• Section: 3.2, 3.3
Introduction
▪ A complex function 𝑤 = 𝑓 (𝑧), is said to be analytic in a
domain 𝑫 if it differentiable at all points in 𝐷.
▪ In order to find out if a complex function is analytic, it is not
possible to check differentiability at each point. Therefore,
an analytic criteria is required to establish analyticity of a
complex function.
▪ This criteria was first used by Cauchy and later it was
formally formulated by Riemann which now known as
Cauchy-Riemann equations.
Augustin-Louis Cauchy
Bernhard Riemann
Cauchy-Reimann Equations; Criterion for
Differentiability
Suppose the real functions 𝑢(𝑥, 𝑦) and 𝑣(𝑥, 𝑦) are continuous and
have continuous first-order partial derivatives in a domain 𝐷. If 𝑢 and 𝑣
satisfy the Cauchy-Riemann equations (CR-equations):
𝜕𝑢 𝜕𝑣
𝜕𝑢
𝜕𝑣
=
and
=−
(1)
𝜕𝑥 𝜕𝑦
𝜕𝑦
𝜕𝑥
at all points of 𝐷, then the complex function 𝑓 𝑧 = 𝑢 𝑥, 𝑦 + 𝑖𝑣(𝑥, 𝑦)
is analytic in 𝐷.
Example 1
i. Determine if the function 𝑓(𝑧) = 𝑧 2 is analytic or not?
1
ii. Determine the points where function 𝑓(𝑧) =
is not
𝑧−1
analytic.
Criterion for Non-Analyticity
If the CR-equations are not satisfied at every point 𝑧 in a
domain 𝐷, then the function 𝑓 𝑧 = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦)
cannot be analytic in 𝐷.
Example 2; Criterion for Non-Analyticity
Cauchy-Reimann Equations in Polar
Coordinates
For complicated complex functions it is better to use
Euler’s formula which require that the CR-equations are
obtained in (𝑟, 𝜃).
The Cauchy-Riemann equations in polar coordinates are
given by:
𝜕𝑢 1 𝜕𝑣
𝜕𝑣
1 𝜕𝑢
=
and
=−
𝜕𝑟 𝑟 𝜕𝜃
𝜕𝑟
𝑟 𝜕𝜃
Example 3; CR-Equations in Polar Coordinates
Sufficient Conditions for Differentiability
If the real functions 𝑢(𝑥, 𝑦) and 𝑣(𝑥, 𝑦) are continuous and have continuous first
order partial derivatives in some neighborhood of a point 𝑧 = 𝑥 + 𝑖𝑦, and if 𝑢 and
𝑣 satisfy the CR-equations at 𝑧, then the function 𝑓 𝑧 = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦) is
differentiable at 𝑧 and 𝑓 ′(𝑧) is given by:
𝜕𝑢
𝜕𝑣
𝜕𝑣
𝜕𝑢
𝑓′(𝑧) =
+𝑖
=
−𝑖
.
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑦
The polar version of the above equation at a point 𝑧 whose polar coordinates are
(𝑟, 𝜃) is then:
𝜕𝑢
𝜕𝑣
1 −𝑖𝜃 𝜕𝑣
𝜕𝑢
−𝑖𝜃
𝑓 ′(𝑧) = 𝑒
+𝑖
= 𝑒
−𝑖
𝜕𝑟
𝜕𝑟
𝑟
𝜕𝜃
𝜕𝜃
Example 4
Determine if the following complex function are analytic and if
they are analytic determine 𝑓′(𝑧) by using the partial
derivatives of real and imaginary parts of 𝑓(𝑧):
i. 𝑓(𝑧) = 𝑒 𝑧
ii. 𝑓 𝑧 = 𝑒 𝑧ҧ
iii. 𝑓 𝑧 = sin 𝑧
iv. 𝑓 𝑧 = 𝑧 6
v. 𝑓(𝑧) = 𝑧 −2
Remark
❑If a complex function 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦) is analytic at
a point 𝑧, then all the derivatives of
𝑓: 𝑓′(𝑧), 𝑓′′(𝑧), 𝑓′′′(𝑧), … are also analytic at 𝑧.
❑As a consequence of this remarkable fact, we can conclude
that all partial derivatives of the real functions 𝑢(𝑥, 𝑦) and
𝑣(𝑥, 𝑦) are continuous at 𝑧.
Laplace Equation
❑ From the continuity of the partial derivatives, we then know that the
second-order mixed partial derivatives are equal. This last fact,
coupled with the Cauchy-Riemann equations, demonstrates that there
is a connection between the real and imaginary parts of an analytic
function 𝑓(𝑧) and the second-order partial differential equation
𝛻 2 𝜙 = 𝜙𝑥𝑥 + 𝜙𝑦𝑦 = 0,
(2)
where 𝛻 2 𝜙 is called the Laplacian of 𝝓.
❑ Equation (2), one of the most famous PDE in Applied Mathematics and
Physics, is known as Laplace’s equation in two variables. It occurs in
gravitation, electrostatics, fluid flow, heat conduction, and other
applications.
Pierre-Simon Laplace
Harmonic Function
A real-valued function 𝜙 of two real variables 𝑥 and 𝑦
that has continuous first and second-order partial
derivatives in a domain 𝐷 and satisfies Laplace’s
equation is said to be harmonic function in 𝑫.
Theorem; Harmonic Function
Suppose the complex function 𝑓 𝑧 = 𝑢 𝑥, 𝑦 + 𝑖𝑣(𝑥, 𝑦)
is analytic in a domain 𝐷. Then the functions 𝑢(𝑥, 𝑦) and
𝑣(𝑥, 𝑦) are harmonic in 𝐷.
Example 5
The function 𝑓(𝑧) = 𝑧 2 = (𝑥 2 − 𝑦 2 ) + 𝑖(2𝑥𝑦) is analytic
everywhere. The functions 𝑢(𝑥, 𝑦) = 𝑥 2 − 𝑦 2 and 𝑣(𝑥, 𝑦) = 2𝑥𝑦
are necessarily harmonic in any domain 𝐷 of the complex plane.
Harmonic Conjugate
Suppose 𝑢(𝑥, 𝑦) is a given real function that is known to be
harmonic in 𝐷. If it is possible to find another real harmonic
function 𝑣(𝑥, 𝑦) so that 𝑢 and 𝑣 satisfy the Cauchy-Riemann
equations throughout the domain 𝐷, then the function 𝑣(𝑥, 𝑦) is
called a harmonic conjugate of 𝑢(𝑥, 𝑦).
By combining the functions as 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦) we obtain a
function that is analytic in 𝐷.
Example 6
Verify that the function 𝑢 𝑥, 𝑦 = 𝑥 3 − 3𝑥𝑦 2 − 5𝑦 is
harmonic in the entire complex plane. Moreover, determine
the harmonic conjugate function of 𝑢.
Note
If 𝑣(𝑥, 𝑦) is a harmonic conjugate of 𝑢(𝑥, 𝑦) in some domain 𝐷, then 𝑢(𝑥, 𝑦) is, in
general, not a harmonic conjugate of 𝑣(𝑥, 𝑦).
Example: The function 𝑓(𝑧) = 𝑧 2 = (𝑥 2 − 𝑦 2 ) + 𝑖(2𝑥𝑦) is analytic everywhere.
The functions 𝑢(𝑥, 𝑦) = 𝑥 2 − 𝑦 2 and 𝑣(𝑥, 𝑦) = 2𝑥𝑦 are harmonic in any domain 𝐷
of the complex plane. In this case 𝑣(𝑥, 𝑦) is a harmonic conjugate of 𝑢(𝑥, 𝑦)
throughout the plane but 𝑢(𝑥, 𝑦) cannot be a harmonic conjugate of 𝑣(𝑥, 𝑦) since
the function:
𝑓 𝑧 = 𝑧 2 = 2𝑥𝑦 + 𝑖(𝑥 2 − 𝑦 2 )
is not analytic anywhere. (Verify!)
Example 7
1. Find the harmonic conjugates of the following:
i. 𝑢 𝑥, 𝑦 = 𝑒 2𝑥 sin 2𝑦 .
ii. 𝑢 𝑥, 𝑦 = 𝑥 2 − 𝑦 2 .
2. Find the value of 𝑘 for which 𝑢(𝑥, 𝑦) = 𝑒 𝑘𝑥 sin 4𝑦 satisfy Laplace’s equation.
Practice Questions
A First Course in
•
Chapter:
3
Complex Analysis with
Applications by Dennis • Exercise: 3.2 Questions: 1-26
G. Zill and Patrick D. • Exercise: 3.3 Questions: 1-20
Shanahan.