Its applied mathematics notes for B.Tech students of Second Semester of Indian Universities.
Strictly according to the syllabus of P.T.U (Jalandhar)
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ENGINEERING MATHEMATICS
B.Tech,Semester-II
RANJIT TUTORIALS
ESTD.2000
TM
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By
Ranjit Singh
Alumnus of Panjab University,Chandigarh
Associate Professor
CGC Technical Campus,Jhanjeri(Mohali)
Department of Applied Sciences
Ex-Head
Department of Applied Sciences
GGS College of Modern Technology,Kharar(Mohali)
rt
ORDINARY DIFFERENTIAL EQUATIONS
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FOURIER SERIES
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ORDINARY
DIFFERENTIAL EQUATIONS
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1. Exact differential equation
2. Equations reducible to exact form by integrating factors
01
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
3. Equations of first order & higher degree
4. Clairaut's Equation
5. Linear differential equation of first order
02
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
6. Linear differential equations of second and higher order
03
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
7. Method of variation of parameters to find P.I.
04
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ORDINARY DIFFERENTIAL EQUATIONS
05
8. Legendre's linear equation & Cauchy's homogeneous
equation
WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
o6
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
07
(
)
)
(
RANJIT TUTORIALS
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
08
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
09
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
10
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
11
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
12
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
13
(
)
(
(
(
)
)
)
(
)
(
)
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
14
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
15
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
Expected Questions
16
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
17
(
(
)
)
(
)
(
)
(
)
(
)
(
)
RANJIT TUTORIALS
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RANJIT TUTORIALS
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
18
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RANJIT TUTORIALS
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
19
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RANJIT TUTORIALS
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
20
RANJIT TUTORIALS
ESTD.2000
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Excellent Coaching for all Classes/Subjects
rt
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RANJIT TUTORIALS
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RANJIT TUTORIALS
ORDINARY DIFFERENTIAL EQUATIONS
Problems for meritorious students
21
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ORDINARY DIFFERENTIAL EQUATIONS
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APPLICATIONS OF ODE
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FOURIER SERIES
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APPLICATIONS OF ODE
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Mean Position
Extreme Position
Extreme Position
o
P
x
a
1. Simple Harmonic Motion (S.H.M)
2. Electric LCR Circuits
L
C
R
E=kcosnt
i
23
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APPLICATIONS OF ODE
EXPECTED QUESTIONS
24
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APPLICATIONS OF ODE
25
MATRICES
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FOURIER SERIES
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MATRICES
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2. Rank of a matrix
(3)
1. Elementary Transformations
3. Normal form of a matrix
26
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MATRICES
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4. Consistency and Solution of linear equations
5. Linearly dependent & Linearly independent vectors
27
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7. Cayley Hamilton Theorem (CHT)
8. Quadratic Form
6. Eigen Values & Eigen Vectors
28
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9. Complex matrices
10. Orthogonal matrix
vely
( )
( )
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11. Unitary matrix
12. Similar matrices
13. Linear transformation
14. Orthogonal transformation
30
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MATRICES
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MATRICES
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MATRICES
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RANJIT TUTORIALS
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MATRICES
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RANJIT TUTORIALS
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MATRICES
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RANJIT TUTORIALS
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MATRICES
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RANJIT TUTORIALS
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MATRICES
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RANJIT TUTORIALS
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MATRICES
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Expected Questions
38
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RANJIT TUTORIALS
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RANJIT TUTORIALS
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MATRICES
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RANJIT TUTORIALS
41
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MATRICES
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RANJIT TUTORIALS
Problems for meritorious students
42
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MATRICES
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RANJIT TUTORIALS
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WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...
...
...
...
...
...
...
...
43
INFINITE SERIES
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FOURIER SERIES
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INFINITE SERIES
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1. Infinite Series
2. Convergence,Divergence & Oscillation of an Infinite Series
3. Auxiliary Series
44
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INFINITE SERIES
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FOURIER SERIES
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RANJIT TUTORIALS
INFINITE SERIES
ALTERNATING SERIES
POSITIVE TERM SERIES
CONVERGENT
CONVERGENT
DIVERGENT
OSCILLATORY
4. Tests of Convergence for positive term series
45
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INFINITE SERIES
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RANJIT TUTORIALS
6
7
8
46
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INFINITE SERIES
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FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
5. Leibnitz's test for convergence of alternating series
6. Absolute and conditional convergence of an alternating
series
7. Important Limits
47
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
48
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
49
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
50
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
51
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
52
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
53
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
54
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Expected Questions
55
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
56
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
57
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Problems for meritorious students
58
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
INFINITE SERIES
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
59
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
rt
COMPLEX NUMBERS
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
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60
1. De-Moivre's Theorem
2. Properties of cis
3. Some important cis forms
4. Values of (cos +isin ) , where p,q are +ve coprime integers
p/q
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
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RANJIT TUTORIALS
61
5. Euler's Formula
6. Circular Functions
7. Hyperbolic Functions
8. Relation between Hyperbolic & Circular Functions
9. Inverse Hyperbolic Functions
10. Logarithm of Complex Numbers
(
)
(
)
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
62
11. Some Standard Series
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
63
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
64
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
65
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
66
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
67
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
68
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
69
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
WRITE YOUR OWN NOTES HERE
rt
...
...
...
...
...
...
...
...
70
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Expected Questions
71
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
72
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
73
(
)
(
)
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
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RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
74
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
75
RANJIT TUTORIALS
ESTD.2000
TM
Excellent Coaching for all Classes/Subjects
rt
COMPLEX NUMBERS
All Rights Reserved
RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
www.ranjittutorials.com,Email: ranjit1516@yahoo.com
RANJIT TUTORIALS
Problems for meritorious students
76
IMPORTANT FORMULAE
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FOURIER SERIES
rt
IMPORTANT BASIC
FORMULAE & RESULTS
RANJIT TUTORIALS
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1. Differentiation
77
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RANJIT TUTORIALS
FOURIER SERIES
RANJIT TUTORIALS
Regd. Office.
# 42,Ranjit Nagar,Near-Bus Stand,Kharar,Distt. Mohali(Punjab),Ph.9876313376
Website:
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RANJIT TUTORIALS
IMPORTANT FORMULAE
2. Integration
78
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
-
Upload your own papers! Earn money and win an iPhone X. -
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Upload your own papers! Earn money and win an iPhone X.