Engineering Mathematics 2

Engineering Mathematics 2

Lecture 6 Yong Sung Park

13.1 Complex numbers

• Complex number is an ordered pair 𝑧 = 𝑥, 𝑦 , where 𝑅𝑒 𝑍 = 𝑥, 𝐼𝑚 𝑍 = 𝑦.

• Imaginary unit: 𝑖 = (0, 1), 𝑖² = −1

• In practice, 𝑧 = 𝑥 + 𝑖𝑦.

• Verify the following:

𝑧₁ ± 𝑧₂ = 𝑥₁ ± 𝑥₂ + 𝑖 𝑦₁ + 𝑦₂

𝑧₁𝑧₂ = 𝑥₁𝑥₂ − 𝑦₁𝑦₂ + 𝑖 𝑥₁𝑦₂ − 𝑥₂𝑦₁ 𝑧₁

𝑧₂ = 𝑥₁𝑥₂ + 𝑦₁𝑦₂ + 𝑖 𝑥₂𝑦₁ − 𝑥₁𝑦₂ 𝑥₂² + 𝑦₂²

• Complex plane

• The complex conjugate number of 𝑧 = 𝑥 + 𝑖𝑦: 𝑧 = 𝑥 − 𝑖𝑦ҧ 𝑧 ҧ𝑧 = 𝑥² + 𝑦²

𝑅𝑒 𝑍 = 𝑥 = 𝑧 + ҧ𝑧

2 , 𝐼𝑚 𝑍 = 𝑦 = 𝑧 + ҧ𝑧 2

x

y z = x + iy

13.2 Polar form of complex numbers.

• With 𝑥 = 𝑟 cos 𝜃, 𝑦 = 𝑟 sin 𝜃, the polar form of a complex number is 𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃

in which

𝑟 = 𝑧 = 𝑥² + 𝑦² = 𝑧 ҧ𝑧 is the absolute value or modulus, and

𝜃 = arg 𝑧 = tan^{−1 𝑦}

𝑥 is the argument, which is multivalued. The principal value of the argument is −𝜋 < Arg 𝑧 ≤ 𝜋 .

x

y z = x + iy

r 𝜃

• Question:

Which is greater? 2𝑖 vs. 5𝑖

• Triangle inequality

𝑧₁ + 𝑧₂ ≤ 𝑧₁ + 𝑧₂

𝑧₁ + 𝑧₂ + ⋯ + 𝑧_𝑛 ≤ 𝑧₁ + 𝑧₂ + ⋯ + 𝑧_𝑛

• Example 2:

Verify the triangle inequality with

𝑧₁ = 1 + 𝑖 and 𝑧₂ = −2 + 3𝑖

• 𝑧₁𝑧₂ = 𝑟₁𝑟₂ cos 𝜃₁ + 𝜃₂ + 𝑖 sin 𝜃₁ + 𝜃₂ , therefore 𝑧₁𝑧₂ = 𝑧₁ 𝑧₂ and arg 𝑧₁𝑧₂ = arg 𝑧₁ + arg 𝑧₂

• ^𝑧1

𝑧₂ = ^𝑟1

𝑟₂ cos 𝜃₁ − 𝜃₂ + 𝑖 sin 𝜃₁ − 𝜃₂ , therefore

𝑧₁

𝑧₂ = ^𝑧1

𝑧₂ and arg ^𝑧1

𝑧₂ = arg 𝑧₁ − arg 𝑧₂

• By induction, 𝑧^𝑛 = 𝑟^𝑛 cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃 , 𝑛 = ⋯ , −2, −1, 0, 1, ⋯

• De Moivre’s formula,

cos 𝜃 + 𝑖 sin 𝜃 ^𝑛 = cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃

• Example:

Express cos 3𝜃 and sin 3𝜃 in terms of cos 𝜃, sin 𝜃 and their powers.

• Roots:

Consider 𝑧 = 𝑤^𝑛, there are 𝑛 values for 𝑤 = ^𝑛 𝑧. Find them.

(hint: use 𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃 and 𝑤 = 𝑅 cos 𝜙 + 𝑖 sin 𝜙 )

• Example:

Find ^𝑛 1 for 𝑛 = 2 and 3.

13.3 Derivative. Analytic function.

• A circle of radius 𝜌 and centre 𝑎: 𝑧 − 𝑎 = 𝜌

•

𝑧 − 𝑎 < 𝜌: open circular disk → 𝜌 -neighbourhood of 𝑎 𝑧 − 𝑎 ≤ 𝜌: closed circular disk

𝑧 − 𝑎 > 𝜌: exterior of circular disk 𝜌₁ < 𝑧 − 𝑎 < 𝜌₂: open annulus

𝜌₁ ≤ 𝑧 − 𝑎 ≤ 𝜌₂: closed annulus

𝑎 𝜌

• 𝑦 > 0: upper half plane 𝑦 < 0: lower half plane 𝑥 > 0: right half plane 𝑥 < 0: left half plane

• Domain is an open connected set.

• A boundary point of a set 𝑆 is a point with its every nbhd contains points in 𝑆 and 𝑆^𝑐.

• Complex function

𝑤 = 𝑓 𝑧 = 𝑢 𝑥, 𝑦 + 𝑖𝑣 𝑥, 𝑦

• Example:

For 𝑤 = ¹

𝑧 𝑧 ≠ 0 , find 𝑢 and 𝑣.

• 𝑓(𝑧) has a limit 𝑙 as 𝑧 → 𝑧₀ from any direction, that is,

𝑧→𝑧lim₀ 𝑓(𝑧) = 𝑙

if for any 𝜖 > 0, there exists 𝛿 > 0 such that 𝑓 𝑧 − 𝑙 < 𝜖 for all 𝑧 ≠ 𝑧₀ and 𝑧 − 𝑧₀ < 𝛿.

• 𝑓(𝑧) is continuous at 𝑧 = 𝑧₀ if lim

𝑧→𝑧₀ 𝑓(𝑧) = 𝑓(𝑧₀).

• A complex function is differentiable at 𝑧 = 𝑧₀ if 𝑓^′ 𝑧₀ = lim

𝑧→𝑧₀

𝑓 𝑧 −𝑓(𝑧₀) 𝑧−𝑧₀

exists. Note the limit is taken from any direction.

• 𝑓′ 𝑧 is called the derivative of 𝑓 𝑧

• 𝑓 𝑧 is called analytic (or holomorphic) in some domain if it is differentiable in the domain.

• The differentiation rules are the same as those for the real functions.

• Example 4: Show that 𝑓 𝑧 = ҧ𝑧 is not analytic.

• Example: Show that

𝑓 𝑧 = 𝑧^𝑛 is analytic in complex plane.

13.4 Cauchy-Riemann equations. Laplace equation

• Consider necessary conditions for 𝑓 𝑧 = 𝑢 𝑥, 𝑦 + 𝑖𝑣 𝑥, 𝑦 to be analytic in nbhd of 𝑧₀ 𝑥₀, 𝑦₀ , by taking the limit

(1) along the horizontal direction; and (2) along the vertical direction.

• 𝑓 is analytic in domain D iff 𝑢 and 𝑣 satisfy Cauchy-Riemann equations:

𝜕𝑢

𝜕𝑥 = ^𝜕𝑣

𝜕𝑦 and ^𝜕𝑣

𝜕𝑥 = − ^𝜕𝑢

𝜕𝑦

• Example 1: Show that

𝑓 𝑧 = ҧ𝑧 does not satisfy C-R equations

• Example 2: Is 𝑓 𝑧 = 𝑒^𝑥 cos 𝑦 + 𝑖 sin 𝑦 analytic?

• Show that, if 𝑓 𝑧 is analytic in a domain D, then both 𝑢 and 𝑣 satisfy 2D Laplace’s equation, respectively.

• 𝑣 is called a harmonic conjugate function of 𝑢 in D and vice versa.

• Example:

Find a harmonic conjugate function for 𝑢 = 𝑥² − 𝑦².