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Algebra, Analysis, Topology & Applied Math Key

This document contains summaries of key concepts from various areas of mathematics, including algebra, analysis, topology, applied mathematics, and miscellaneous topics. It covers topics such as complex numbers, limits, continuity, differentiation, integration, singular points, factorials, logarithms, and solutions to equations. The document provides high-level overviews of important mathematical definitions, theorems, and methods across multiple domains.

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122 views1 page

Algebra, Analysis, Topology & Applied Math Key

This document contains summaries of key concepts from various areas of mathematics, including algebra, analysis, topology, applied mathematics, and miscellaneous topics. It covers topics such as complex numbers, limits, continuity, differentiation, integration, singular points, factorials, logarithms, and solutions to equations. The document provides high-level overviews of important mathematical definitions, theorems, and methods across multiple domains.

Uploaded by

siva
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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KEY Section 4: Applied Mathematics

Section 1: Algebra
4.1 2
1.1 9 λ = (2n + 1)2 π4 ,
1.2 b,c u = C sin(2n + 1) π2 , n = 0, 1, 2, · · ·
1.3 The multiplicative group {−1, 1}
1.4 a,c 4.2
−2
1.5 a − 2b − c = 0 c=
b−a
1.6  
1 3 4
 1 0 2  4.3
π

 0 1

0  √
2
1
7
0 0 3
4.4
1.7 0 x2 − x
y(x) =
1.8 a,b,c 4
1.9 a √
1.10 λ2 − 3 2λ + 4 4.5 L(y 00 )(s) = s2 L(y)(s) − sy(0) − y 0 (0)
4.6 x = 0, irregular singular point, x = 1, regular
Section 2: Analysis singular point
2.1 sin 1 4.7 0
2.2 4 4.8 y(x) = x3 + C1 x2 + C2
e
2.3 b,c 4.9
2.4 a,b Z b
(b − a)
2.5 R\{−1} f (x) dx ∼ [f (a) + 4f ((a + b)/2)) + f (b)]
a 6
2.6 b
2.7 4.10 3
Z 1
0
|f (t)| dt Section 5: Miscellaneous
0

5.1
2.8 m!(m + 1)!
∞ 2n
X
n−1 x
(−1) (m − n + 1)!
n=1
(2n − 1)(2n)
5.2 2n−1 (3n + 2)
(4n+1)π
2.9 e , n∈Z 5.3 a,b
2.10 {z = x + iy : y ≥ 0} 5.4 log N
5.5 0
Section 3: Topology 3 2
5.6 √ − 16x + 64x − 9 = 0
x
3.1 b 5.7 37
3.2 a,c 5.8
a + 3b
3.3 B and D are homeomorphic
3.4 a,b,c 4
3.5 a,b
3.6 a,b 5.9 a,b
3.7 b 5.10 a,b
3.8 a,b
3.9 c
3.10 b,c

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