1. 1 .INTRODUCTION
It’s a virtually impossible task to do justice, in a short span of time and space, to the
great genius of LEONHARD EULER. All we can do in this project, is to bring across some
glimpses of Euler’s incredibly voluminous and diverse work. Leonhard Euler was an 18th
century physicist and scholar who was responsible for developing many concepts that are an
integral of modern mathematics. To begin with his a brief outline of Euler’s Life.
1.1 His Life
Euler Leonard is born in 15th April 1707 in Basel, Switzerland and passed away
during 18th September 1783 due to Cerebral hemorrhage in St Peterburg, Russia. He is
the son of a Lutheran minister in Basel. He was educated by the renown Johann
Bernoulli and befriended with 2 of his sons Nicholas and Daniel who have befriended
with Euler throughout his life time. Euler is the elder’s among the six children in his
family, but have no intention to follow his father’s footstep as a minister. He entered the
University of Basel when he is just 13 years old and received his Master’s Degree at the
age of 16 years old. His studies are not solely on mathematics only, he also studies into
the field of astronomy, languages, physics and medicine.
Euler have his life tough as he have completely lost sight in his right eye, most
probably due to observing the sun with naked eyes and he states that due to that he will be
having less distraction. At the age of 19 he published his first formal mathematical paper
about ship’s mast optimum placement, then he leaves to St. Petersburg Academy during

2. which is located Russia during the year 1727. During the year 1741, Euler leaves
Petersburg and takes a postion under Frederick the Great in Berlin Academy but returned
to St. Peterburg eventually when the Catherin the Great reigns.
At the age of 50, Euler turns completely blind as by then his left eye are blind due to
cataract, but even so, Euler was highly stated that even when he is blind he is still able to
do his incredible mathematical calculation.
1.2 Euler Leonard’s facts
 Euler introduced some of the mathematical terminology and notation that are still use
today, especially in mathematical Analysis.
 He is also widely remembered for his contribution in mechanics, fluid dynamics,
optics, astronomy and music.
 Euler is remembered as the most important mathematician of the 18th century, as well
as one of the most and greatest mathematician who ever lived.
 Pierre-Simon Laplace expressed Euler’s important to mathematics, Quoting Laplace
with his saying: ‘Read Euler, read Euler, he is the master of us all.’
 Euler’s father was a friend to Johann Bernoulli, and Bernoulli have become Euler’s
most influential person throughout his life.
 When Euler was still being tutored under Bernoulli in terms of mathematics.
 Newton and Descartes philosophies was on comparison with Euler’s Master’s Thesis.
 Euler made many signification contribution in math in terms of Infinitesimal
Calculus, Geometry, Trigonometry, Number Theory and Algebra.

3.  Euler wrote many ideas in lunar theory , continuum physics, and other areas.
 There are 2 things in mathematics that are named after Euler, which is the Euler’s
number ‘e’ exponential which have an approximate value of 2.7183, which the next
thing is Euler’s constant which is known as the Euler-Mascheroni Constant γ
(gamma), which have an approximated value of 0.5772.
2.0 MATHEMATICAL NOTATION CREATED BY LEONHARD EULER
𝑒 The base of the natural Logarithm, a constant equal to 2.71828
𝑖 The “imaginary Unit” ,equal to the square root of -1
𝑓(𝑥) The function 𝑓 as applied to the variable or argument 𝑥
∑ Sigma ,the sum or total of a set of number
𝑎, 𝑏, 𝑐
𝑥, 𝑦, 𝑧
𝑎, 𝑏, 𝑐 are constants such as the sides of a triangle , 𝑥, 𝑦, 𝑧 are
variable or unknowns in an equation
𝑠𝑖𝑛, 𝑐𝑜𝑠, 𝑡𝑎𝑛, 𝑐𝑜𝑡, 𝑠𝑒𝑐, 𝑐𝑠𝑐 Trigonometric functions for sine,
cosine ,tangent ,cotangent ,secant ,cosecant
𝜋 Pi, the ratio of a circle’s circumference to its diameter

4. 3.0 APPLICATION OF EULER’S FORMULA
3.1) Complex Impedance in Phasor Analysis in RLC Circuit
In an A/C circuit with a sinusoidal supply of voltage, we can say that:
𝑉( 𝑡) = 𝑉𝑜 cos(𝑤𝑡 + ∅)
However, if there are presence of capacitors and inductors which produce ∓90° changes in
phase. Therefore, Euler’s formula can be used to add imaginary part to voltage or current:
𝑉( 𝑡) = 𝑉𝑜 cos(𝑤𝑡 + ∅) + 𝑗 𝑉𝑜cos( 𝑤𝑡 + ∅) = 𝑉𝑜 𝑒 𝑗(𝑤𝑡+∅)
This works only for linear circuit, which is one that contains only passive component such as
resistors, capacitors and inductors and also voltage or current source. Voltage for resistors
is 𝑣 = 𝐼𝑅 sin 𝑤𝑡; while voltage for capacitor is 𝑣 =
𝐼
𝑊𝐶
sin(𝑤𝑡 −
1
2
𝜋) and the voltage for
inductor is 𝑣 = 𝐼𝑤𝐿 sin(𝑤𝑡 +
1
2
𝜋).
From the Euler’s formula, we know that sin 𝜃 = Im (𝑒 𝑗𝜃
):
V=
{
Im(IR𝑒 𝑗𝑤𝑡
)
Im(
I
WC
𝑒 𝑗( 𝑤𝑡−
𝜋
2
)
)
Im(wLI𝑒
𝑗( 𝑤𝑡+
𝜋
2
)
)
Since 𝑒
𝑗(
𝜋
2
)
= 𝑗 while 𝑒
−𝑗(
𝜋
2
)
= −𝑗, hence:
v = Im(IZ𝑒 𝑗𝑤𝑡
)

5. Where
Z={
R
−
j
WC
jwL
Z is known as complex impedance where | 𝑍| = [𝑅2
+ (𝐿𝑤 −
1
𝐶𝑊
)
2
]
1/2
is the impedance of
the RLC circuit while ∅ = 𝑡𝑎𝑛 −1
(
𝐿𝑤−
1
𝐶𝑤
𝑅
) is the phase of the RLC circuit.
Therefore, by using phasor analysis on RLC circuit, we actually can find out the current flow
in the circuit and the voltage required for each passive component. We need to analysis the
value required so that we can design the circuit accordingly.
3.2) Fourier transform
Fourier transform is a powerful mathematical tool that allows you to view your signals in a
different domain, inside which several difficult problems become very simple to analyze.
Fourier series is an expansion of a periodic function 𝑓(𝑡) of period 𝑇 = 2𝜋/𝑤 and the base
set is the set of sine function:
𝐹( 𝑡) = 𝐴0 + 𝐴1 sin(𝑤𝑡 + ∅1) + 𝐴2 sin(2𝑤𝑡 + ∅2) + ⋯+ 𝐴 𝑛 sin(𝑛𝑤𝑡 + ∅ 𝑛) + ⋯
Where 𝐴 𝑛 and ∅ 𝑛 are constants, 𝑤 =
2𝜋
𝑇
𝑎𝑛𝑑 ∅ 𝑛 is its phase angle
(Resistor)
(Capacitor)
(Inductor)

6. 𝐴 𝑛 sin(𝑛𝑤𝑡 + ∅ 𝑛 ) ≡ (𝐴 𝑛 cos∅ 𝑛) sin 𝑛𝑤𝑡 +(𝐴 𝑛 sin ∅ 𝑛)cos 𝑛𝑤𝑡
≡ 𝑏 𝑛 sin 𝑛𝑤𝑡 + 𝑎 𝑛 cos 𝑛𝑤𝑡
Let
𝑏 𝑛 = 𝐴 𝑛 cos∅ 𝑛 𝑎 𝑛 = 𝐴 𝑛 sin ∅ 𝑛
Hence
𝑓( 𝑡) =
1
2
𝑎0 + ∑ 𝑎 𝑛 cos 𝑛𝑤𝑡 + ∑ 𝑏 𝑛 sin 𝑛𝑤𝑡
∞
𝑛=1
∞
𝑛=1
The equation above is the Fourier series expansion of the function 𝑓( 𝑡) and 𝑎 𝑛 𝑎𝑛𝑑 𝑏 𝑛 are
the Fourier coefficient. Fourier coefficient actually can be related to the use of the phasor
notation 𝑒 𝑗𝑛𝑤𝑡
= cos 𝑛𝑤𝑡 + 𝑗sin 𝑛𝑤𝑡.
After a series of integration, the 𝑎 𝑛 𝑎𝑛𝑑 𝑏 𝑛 are transformed into Euler’s formulae:
𝑎 𝑛 =
2
𝑇
∫ 𝑓( 𝑡) cos 𝑚𝑤𝑡 𝑑𝑡 (𝑛 = 0,1,2, … )
𝑑+𝑇
𝑑
𝑏 𝑛 =
2
𝑇
∫ 𝑓( 𝑡)sin 𝑚𝑤𝑡 𝑑𝑡 (𝑛 = 1,2,3, …)
𝑑+𝑇
𝑑
We know that
𝑒 𝑗𝑛𝑤𝑡
= cos 𝑛𝑤𝑡 + 𝑗sin 𝑛𝑤𝑡.
Hence,
𝑎 𝑛 + 𝑗𝑏 𝑛 =
2
𝑇
∫ 𝑓( 𝑡) 𝑒 𝑗𝑛𝑤𝑡
𝑑𝑡
𝑑+𝑇
𝑑

7. Fourier transform is vital for solving some of the partial differential equation such as heat
equation. Besides, Fourier transform is applicable in nuclear magnetic resonance (NMR),
magnetic resonance imaging (MRI) and mass spectrometry. We can conclude that Fourier
transform is useful for the scientific analysis by comparing the data.
3.3) De Moivre’s Theorem
From Euler’s formula, 𝑒 𝑗𝜃
= 𝑐𝑜𝑠𝜃 + 𝑗𝑠𝑖𝑛𝜃 , hence 𝑧 = 𝑟𝑒 𝑗𝜃
. When z to the power of n,
according to index law, 𝑧 𝑛
= 𝑟 𝑛
(𝑒 𝑗𝜃
) 𝑛
= 𝑟 𝑛
(𝑒 𝑗( 𝑛𝜃)
). If this is translated into polar form,
𝑧 𝑛
= 𝑟 𝑛
(cos 𝑛𝜃 + 𝑗sin 𝑛𝜃 ). This result is known as De Moivre’s Theorem.
Example)
Evaluate (1 − 𝑗)10
using De Moivre’s Theorem.
Solution:
First, express (1-j) in polar form.
Let (1 − 𝑗) = 𝑧
|z|=|1 − 𝑗| = √12+(−1)2 = √2 and arg (z) = arg (1 − 𝑗) =
− tan−1
(
1
1
)=−
1
4
𝜋
𝑧 = 1 − 𝑗 = √2 [cos(−
1
4
𝜋) + 𝑗 sin (−
1
4
𝜋)]
= √2 (cos
1
4
𝜋 − 𝑗 sin
1
4
𝜋)

8. By using De Moivre’s Theorem,
(1 − 𝑗)10
= (√2 )10
[cos(
10
4
𝜋) − 𝑗 sin (
10
4
𝜋)]
= 25
[cos(
5
2
𝜋) − 𝑗 sin (
5
2
𝜋)]
= 25
[cos(
1
2
𝜋) − 𝑗 sin (
1
2
𝜋)]
= 25(0 − 𝑗 )
= −32𝑗
De Moivre’s theorem is important because it connects complex numbers and trigonometry
and many application in Mathematics related to complex number and trigonometry.
3.4) Relationship between circular & hyperbolic function
From Euler’s formula, 𝑒 𝑗𝜃
= 𝑐𝑜𝑠𝜃 + 𝑗𝑠𝑖𝑛𝜃. In fact, there is a link between circular and
hyperbolic functions.
𝑒 𝑗𝜃
= 𝑐𝑜𝑠𝜃 + 𝑗𝑠𝑖𝑛𝜃 And 𝑒−𝑗𝜃
= 𝑐𝑜𝑠𝜃 − 𝑗𝑠𝑖𝑛𝜃
Hence,
𝑐𝑜𝑠𝜃 =
𝑒 𝑗𝜃
+𝑒−𝑗𝜃
2
And 𝑠𝑖𝑛𝜃 =
𝑒 𝑗𝜃
−𝑒−𝑗𝜃
2𝑗
According to hyperbolic function,
cosh 𝑥 =
𝑒 𝑥
+𝑒−𝑥
2
And sinh 𝑥 =
𝑒 𝑥
−𝑒−𝑥
2
1
2

9. Comparing Eq.1 and Eq.2:
1) cosh 𝑗𝑥 =
𝑒 𝑗𝑥
+𝑒−𝑗𝑥
2
= cos 𝑥 And 2) sinh 𝑗𝑥 =
𝑒 𝑗𝑥
−𝑒−𝑗𝑥
2𝑗
= j sin x
3) tanh 𝑗𝑥 = 𝑗 tan 𝑥
4) cos 𝑗𝑥 =
𝑒 𝑗2 𝑥
+𝑒−𝑗2 𝑥
2
=
𝑒 𝑥
+𝑒−𝑥
2
= cosh 𝑥 and 5) tan 𝑗𝑥 = 𝑗 tanh 𝑥
6) sin 𝑗𝑥 =
𝑒 𝑗2 𝑥
−𝑒−𝑗2 𝑥
2𝑗
=
𝑒−𝑥
−𝑒 𝑥
2𝑗
= j sinh 𝑥
To summarise:
1) cosh 𝑗𝑥 = cos 𝑥 4) cos 𝑗𝑥 = cosh 𝑥
2) sinh 𝑗𝑥 = j sin x 5) sin 𝑗𝑥 = j sinh 𝑥
3) tanh 𝑗𝑥 = 𝑗 tan 𝑥 6) tan 𝑗𝑥 = 𝑗tanh 𝑥
Example)
Evaluate sinh(3 + 𝑗4).
Solution:
From the trigonometry identity, we know that,
sinh( 𝐴 + 𝐵) = sinh 𝐴 cosh 𝐵 + cosh 𝐴 sinh 𝐵
sinh(3 + 𝑗4) = sinh 3 cosh 𝑗4 + cosh3 sinh 𝑗4
From the Equation that define previously,
sinh(3 + 𝑗4) = sinh3 cos4 + 𝑗 cosh3sin 4
= −6.548 − 𝑗7.619

10. Hyperbolic and circular trigonometry actually have a direct relationship with the Lorentz
transformation and the application to special relativity. Osborn’s theorem can be used to transform
trigonometry identity into hyperbolic identity.
References
 https://en.wikipedia.org/wiki/Euler%27s_formula
 Burton, D. (1998) Elementary Number Theory. St. Louis: The McGraw -Hill Companies, Inc.
 Cooke, R. (1997) the History of Mathematics. New York: John Wiley and Sons, Inc.
 Dunham, W. (1990) Journey through Genius. New York: John Wiley and Sons.
 Price,J., Rath, J.N.,Leschensky, W. (1992) Pre-Algebra,a transition to algebra. Lake Forest:
Macmillan / McGraw - Hill Publishing Company.
 G. Assayag,H.-G. Feichtinger, and J. F. Rodrigues, eds., Mathematics and Music: A Diderot
Mathematical Forum, Springer, Berlin, 2002. [2] P. Bailhache, Deux math´ematiciens
musiciens: Euler et d’Alembert, Physis Riv. Internaz. Storia Sci. (N.S.),32 (1995), pp. 1–35.