MAT 2101 Complex Variables, Laplace and Z transform

Complex Variable, Laplace and Z-Transformation Summer 24-25


Chapter 1
Laplace Transformation
Definition: For all 𝑡 ≥ 0,
∞

𝐹(𝑠) = ∫ 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡.
0
If the resulting integral exists (i.e., has some finite value), it is a function of 𝑠, 𝑠 may be real
or complex.
This function 𝐹(𝑠) of variable 𝑠 is called Laplace Transformation of the original function
𝑓(𝑡) and will be denoted by ℒ{𝑓(𝑡)}, where ℒ denotes the Laplace transform operator. Thus
∞

ℒ{𝑓(𝑡)} = 𝐹(𝑠) = ∫ 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡.
0
The original function 𝑓(𝑡) is called the inverse transform or inverse of 𝐹(𝑠) and will be
denoted byℒ −1 {𝐹(𝑠)}.
∴ 𝑓(𝑡) = ℒ −1 {𝐹(𝑠)}.


Properties of Laplace transformation:

1. ℒ{𝑎𝑓1 (𝑡) + 𝑏𝑓2 (𝑡)} = 𝑎ℒ{𝑓1 (𝑡)} + 𝑏ℒ{𝑓2 (𝑡)} (linearity), where 𝑎 &𝑏 are constants.

2. If ℒ{𝑓(𝑡)} = 𝐹(𝑠), then ℒ{𝑒 𝑎𝑡 𝑓(𝑡)} = 𝐹(𝑠 − 𝑎) (first shifting or translation)
𝑑𝑛
3. If ℒ{𝑓(𝑡)} = 𝐹(𝑠), then ℒ{𝑡 𝑛 𝑓(𝑡)} = (−1)𝑛 𝑑𝑠𝑛 [𝐹(𝑠)] (multiplication by 𝑡 𝑛 )

Proof of some selective formulae:
Example 1.1
1
ℒ{1} = 𝑠 , (𝑠 > 0)
Proof: From the definition of Laplace transformation, we know that,
∞
ℒ{𝑓(𝑡)} = ∫ 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡
0
∞
ℒ{1} = ∫ 𝑒 −𝑠𝑡 1 𝑑𝑡
0
∞
= ∫ 𝑒 −𝑠𝑡 𝑑𝑡
0
𝑒 −𝑠𝑡 ∞
= ( )|
−𝑠 0
1
= 𝑤ℎ𝑒𝑟𝑒 𝑠 > 0
𝑠

Example 1.2
1
ℒ{𝑡} = 𝑠2 , (𝑠 > 0)
Proof: From the definition of Laplace transformation, we know that,
∞
ℒ{𝑓(𝑡)} = ∫ 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡
0

1

, Complex Variable, Laplace and Z-Transformation Summer 24-25

∞
ℒ{𝑡} = ∫ 𝑒 −𝑠𝑡 𝑡 𝑑𝑡
0
𝑒 −𝑠𝑡 ∞ ∞ −𝑠𝑡
𝑒
= 𝑡( )| + ∫ 𝑑𝑡
−𝑠 0 0 𝑠
1 ∞
= 0 + ∫ 𝑒 −𝑠𝑡 𝑑𝑡
𝑠 0
−𝑠𝑡 ∞
𝑒
= |
−𝑠 2 0
1
= 𝑤ℎ𝑒𝑟𝑒 𝑠 > 0
𝑠2

Example 1.3
1
ℒ{𝑒 𝑎𝑡 } = , (𝑠 > 𝑎)
𝑠−𝑎
Proof: From the definition of Laplace transformation, we know that,
∞

ℒ{𝑓(𝑡)} = ∫ 𝑒 −𝑠𝑡 𝑓(𝑡)𝑑𝑡
0
∞

ℒ{𝑒 𝑎𝑡 } = ∫ 𝑒 −𝑠𝑡 𝑒 𝑎𝑡 𝑑𝑡
0
𝑝

= lim ∫ 𝑒 −(𝑠−𝑎)𝑡 𝑑𝑡
𝑝→∞
0
𝑝
𝑒 −(𝑠−𝑎)𝑡
= lim |
𝑝→∞ −(𝑠 − 𝑎)
0
1 − 𝑒 −(𝑠−𝑎)𝑝
= lim
𝑝→∞ (𝑠 − 𝑎)
1
= where 𝑠 > 𝑎
𝑠−𝑎

Example 1.4
𝒔 𝒂
𝓛{𝒄𝒐𝒔 𝒂𝒕} = 𝒔𝟐 +𝒂𝟐, if |𝒔| > 𝒂 and 𝓛{𝒔𝒊𝒏 𝒂𝒕} = 𝒔𝟐 +𝒂𝟐, if |𝒔| > 𝒂
Proof:
1
ℒ{𝑒 𝑖𝑎𝑡 } =
𝑠 − 𝑖𝑎
𝑠 + 𝑖𝑎
=
(𝑠 − 𝑖𝑎)(𝑠 + 𝑖𝑎)
𝑠 + 𝑖𝑎
= 2
𝑠 + 𝑎2
𝑠 𝑎
= 2 2
+𝑖 2
𝑠 +𝑎 𝑠 + 𝑎2
But 𝑒 𝑖𝑎𝑡 = cos 𝑎𝑡 + 𝑖 sin 𝑎𝑡
So,



2

, Complex Variable, Laplace and Z-Transformation Summer 24-25

∞

ℒ{𝑒 𝑖𝑎𝑡 } = ∫ 𝑒 −𝑠𝑡 𝑒 𝑖𝑎𝑡 𝑑𝑡
0
∞

= ∫ 𝑒 −𝑠𝑡 (cos 𝑎𝑡 + 𝑖 sin 𝑎𝑡)𝑑𝑡
0
∞ ∞

= ∫ 𝑒 −𝑠𝑡 cos 𝑎𝑡 𝑑𝑡 + 𝑖 ∫ 𝑒 −𝑠𝑡 sin 𝑎𝑡 𝑑𝑡
0 0
= ℒ{cos 𝑎𝑡} + {sin 𝑎𝑡}
Comparing (1) and (2), we have on equating real and imaginary parts,
𝑠 𝑎
ℒ{cos 𝑎𝑡} = 𝑠2 +𝑎2 and ℒ{sin 𝑎𝑡} = 𝑠2 +𝑎2 if |𝑠| > 𝑎.

Some related important formulae
𝑒 𝑖𝑎𝑡 − 𝑒 −𝑖𝑎𝑡
sin 𝑎𝑡 =
2𝑖
𝑒 + 𝑒 −𝑖𝑎𝑡
𝑖𝑎𝑡
cos 𝑎𝑡 =
2
𝑒 − 𝑒 −𝑎𝑡
𝑎𝑡
sinh 𝑎𝑡 =
2
𝑒 𝑎𝑡 + 𝑒 −𝑎𝑡
cosh 𝑎𝑡 =
2
𝑒 𝑎𝑡 (𝑎 sin 𝑏𝑡 − 𝑏 cos 𝑏𝑡)
∫ 𝑒 𝑎𝑡 sin 𝑏𝑡 𝑑𝑡 =
𝑎2 + 𝑏 2
𝑒 𝑎𝑡 (𝑎 cos 𝑏𝑡 + 𝑏 sin 𝑏𝑡)
∫ 𝑒 𝑎𝑡 cos 𝑏𝑡 𝑑𝑡 =
𝑎2 + 𝑏 2
Table of Some Important Formulae Related to Laplace Transformation:
1 𝑐
1. ℒ{1} = , (𝑠 > 0) 2. ℒ{𝑐} = , 𝑐 is any constant, (𝑠 > 0)
𝑠 𝑠
1 𝑛!
3. ℒ{𝑡} = , (𝑠 > 0) 4. ℒ{𝑡 𝑛 } = , when 𝑛 = 0,1,2,3, . ..
𝑠2 𝑠𝑛+1

1 1
5. ℒ{𝑒 𝑡 } = , (𝑠 > 1) 6. ℒ{𝑒 𝑎𝑡 } = , (𝑠 > 𝑎)
𝑠−1 𝑠−𝑎
𝑎 𝑠
7. ℒ{sin 𝑎𝑡} = , |𝑠| > 𝑎 8. ℒ{cos 𝑎𝑡} = 2 , |𝑠| > 𝑎
𝑠 2 + 𝑎2 𝑠 + 𝑎2
𝑎 𝑠
9. ℒ{sinh 𝑎𝑡} = , |𝑠| > 𝑎 10. ℒ{cosh 𝑎𝑡} = , |𝑠| > 𝑎
𝑠 2 − 𝑎2 𝑠 2 − 𝑎2
𝜔 𝑠−𝑎
11. ℒ{𝑒 𝑎𝑡 sin 𝜔𝑡} = 12. ℒ{𝑒 𝑎𝑡 cos 𝜔𝑡} =
(𝑠 − 𝑎)2 + 𝜔 2 (𝑠 − 𝑎)2 + 𝜔 2
𝜔 𝑠−𝑎
13. ℒ{𝑒 𝑎𝑡 sinh 𝜔𝑡} = 14. ℒ{𝑒 𝑎𝑡 cosh 𝜔𝑡} =
(𝑠 − 𝑎)2 − 𝜔 2 (𝑠 − 𝑎)2 − 𝜔 2
𝑑𝑛
15. ℒ{𝑡 𝑛 𝑓(𝑡)} = (−1)𝑛 [𝐹(𝑠)]
𝑑𝑠 𝑛

3