Engineering Mathematics for GATE ME/CE/EC papers. Section 4 : Complex Variables

1. Section 4 : Complex
Variables
Topic 3 : Cauchy-Riemann Equations

2. Cauchy-Riemann Equations
 Theorem 1 :- Necessary conditions for a function f(z) to be analytic. If 𝑓 𝑧 = 𝑢 𝑥, 𝑦 +
𝑖 𝑣(𝑥, 𝑦) is analytic at 𝑧0, then 𝑢 and 𝑣 satisfy the Cauchy-Riemann equations,
𝜕𝑢
𝜕𝑥
=
𝜕𝑣
𝜕𝑦
𝑖. 𝑒. 𝑢 𝑥 = 𝑣 𝑦
𝜕𝑣
𝜕𝑥
= −
𝜕𝑢
𝜕𝑦
𝑖. 𝑒. 𝑣 𝑥 = −𝑢 𝑦
at every point in some neighbourhood of point 𝑧0 provided 𝑢 𝑥, 𝑢 𝑦, 𝑣 𝑥 & 𝑣 𝑦 exists.

3. Cauchy-Riemann Equations
 Theorem 2 :- Sufficient condition for f(z) to be analytic, If
1) 𝑓 𝑧 = 𝑢 𝑥, 𝑦 + 𝑖 𝑣(𝑥, 𝑦) is defined at every point in some neighbourhood of point 𝑧0.
2) 𝑢 𝑎𝑛𝑑 𝑣 satisfy CR equations 𝑢 𝑥 = 𝑣 𝑦, 𝑣 𝑥 = −𝑢 𝑦 at every point in some neighbourhood of
point 𝑧0.
3) 𝑢, 𝑣, 𝑢 𝑥, 𝑢 𝑦, 𝑣 𝑥, 𝑣 𝑦 are continuous at every point in some neighbourhood of point 𝑧0.
Then the function f(z) is analytic at 𝑧0 & 𝑓′
𝑧 = 𝑢 𝑥 + 𝑖𝑣 𝑥

4. Cauchy-Riemann Equations
 Example 1 :- Analytic or Not?  𝑓 𝑧 = 𝑒 𝑥 cos 𝑦 + 𝑖 𝑒 𝑥 sin 𝑦
Solution :- 𝑓 𝑧 = 𝑢 𝑥, 𝑦 + 𝑖 𝑣(𝑥, 𝑦)  𝑢 = 𝑒 𝑥 cos 𝑦 and 𝑣 = 𝑒 𝑥 sin 𝑦
𝑢 𝑥 =
𝜕𝑢
𝜕𝑥
= 𝑒 𝑥 cos 𝑦 𝑢 𝑦 =
𝜕𝑢
𝜕𝑦
= −𝑒 𝑥 sin 𝑦
𝑣 𝑥 =
𝜕𝑣
𝜕𝑥
= 𝑒 𝑥
sin 𝑦 𝑣 𝑦 =
𝜕𝑢
𝜕𝑦
= 𝑒 𝑥
cos 𝑦
𝑢 𝑥 = 𝑣 𝑦 and 𝑣 𝑥 = −𝑢 𝑦
Ans:- 𝑓 𝑧 is an Analytic function.

5. Cauchy-Riemann Equations
 Example 2 :- If 𝑢 𝑥, 𝑦 = 𝑥2 − 𝑦2 − 𝑦 − 2 is real part of analytic function 𝑓 𝑧 = 𝑢 + 𝑖𝑣, then
find 𝑓(𝑧).
Solution :- 𝑢 𝑥 = 2𝑥 and 𝑢 𝑦 = −𝑣 𝑥 = −2𝑦 − 1  𝑣 𝑥 = 2𝑦 + 1
From theorem 2 𝑓′
𝑧 = 𝑢 𝑥 + 𝑖 𝑣 𝑥 = 2𝑥 + 𝑖 2𝑦 + 1 = 𝟐𝐳 + 𝐢  replacing x with z and y with 0.
𝑓 𝑧 = 2𝑧 𝑑𝑧 + 𝑖 𝑑𝑧 = 2
𝑧2
2
+ 𝑖𝑧 + 𝑐
𝑓 𝑧 = 𝑧2
+ 𝑖𝑧 + 𝑐
where, 𝑐 = 𝑐1 + 𝑖𝑐2

6. Cauchy-Riemann Equations
 Polar form :- variables 𝑟 𝑎𝑛𝑑 𝜃
𝜕𝑢
𝜕𝑟
=
1
𝑟
𝜕𝑣
𝜕𝜃
𝑖. 𝑒. 𝑢 𝑟 =
1
𝑟
𝑣 𝜃
𝜕𝑢
𝜕𝜃
=
𝜕𝑣
𝜕𝑟
𝑖. 𝑒.
−1
𝑟
𝑢 𝜃 = 𝑣𝑟
𝑓′ 𝑧 = (𝑢 𝑟 + 𝑖 𝑣𝑟)𝑒−𝑖𝜃

7. Harmonic Functions
 𝑢 𝑥, 𝑦 →
𝜕2 𝑢
𝜕𝑥2 +
𝜕2 𝑢
𝜕𝑦2 = 𝑢 𝑥𝑥 + 𝑢 𝑦𝑦 = 0 → 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛
 A function 𝑢 𝑥, 𝑦 is called harmonic if 𝑢 𝑥, 𝑢 𝑦, 𝑢 𝑥𝑥 𝑎𝑛𝑑 𝑢 𝑦𝑦 are continuous and the function
satisfies Laplace equation.
 Important Note :-
 If 𝑓 𝑧 = 𝑢 + 𝑖𝑣 is analytic function, then 𝑢 & 𝑣 are harmonic functions.
 If 𝑢 & 𝑣 are harmonic functions, then𝑓 𝑧 = 𝑢 + 𝑖𝑣 may or may not be an analytic function.

8. Harmonic Conjugate Function
 If 𝑢 & 𝑣 are harmonic functions and 𝑢 + 𝑖𝑣 is analytic, then 𝑣 is called harmonic conjugate
function of 𝑢.
 How to find a harmonic conjugate?
 If 𝑣 𝑥, 𝑦 is given to find its H.C. 𝑢 𝑥, 𝑦 then…
 𝑑𝑢 =
𝜕𝑢
𝜕𝑥
𝑑𝑥 +
𝜕𝑢
𝜕𝑦
𝑑𝑦
 𝑑𝑢 = 𝑢 𝑥 𝑑𝑥 + 𝑢 𝑦 𝑑𝑦 = 𝑣 𝑦 𝑑𝑥 − 𝑣 𝑥 𝑑𝑦
 𝑢 = 𝑣 𝑦 𝑑𝑥 𝑡𝑟𝑒𝑎𝑡𝑖𝑛𝑔 𝑦 𝑐𝑜𝑛𝑠𝑡 + −𝑣 𝑥 𝑑𝑦(𝑜𝑛𝑙𝑦 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡𝑒𝑟𝑚𝑠 𝑓𝑟𝑒𝑒 𝑓𝑟𝑜𝑚 𝑥)

9. Harmonic Conjugate Function
 Example :- If 𝑢 = 𝑥3 − 3𝑥𝑦2 + 7, then find its harmonic conjugate.
Solution :- 𝑑𝑣 = 𝑣 𝑥 𝑑𝑥 + 𝑣 𝑦 𝑑𝑦 = −𝑢 𝑦 𝑑𝑥 + 𝑢 𝑥 𝑑𝑦
𝑢 𝑥 = 3𝑥2 − 3𝑦2
𝑢 𝑦 = −6𝑥𝑦
𝑑𝑣 = 6𝑥𝑦 𝑑𝑥 + 3𝑥2 − 3𝑦2 𝑑𝑦
𝑣 = 6𝑥𝑦 𝑑𝑥 + −3𝑦2 𝑑𝑦
𝑣 =
6𝑦𝑥2
2
+
−3𝑦3
3
+ 𝑘 = 3𝑥2
𝑦 − 𝑦3
+ 𝑘
𝑘 = real constant