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Some Important Notes
N. Annamalai M.Sc,M.Phil.,DST- INSPIRE (SRF)
Department of MathematicsBharathidasan University
Tiruchirappalli-24Email: [email protected]
Website: https://annamalaimaths.wordpress.com
October 14, 2017
2
ALGEBRA
1. Let GL(n, q) be the group of all n×n invertible matrices over the finitefield Fq, where q = pm, p is a prime and some positive integer m. Then
(i)O(GL(n, q)) = qn(n−1)
2 (qn − 1)(qn−1 − 1) · · · (q − 1).
(ii) Order of Sylow p-subgroup of GL(n, q) is qn(n−1)
2 .
(iii) O(SL(n, q)) =qn(n−1)
2 (qn − 1)(qn−1 − 1) · · · (q − 1)
(q − 1).
2. The center Z(GL(n, q)) = kIn | k ∈ F∗q, where In is the identitymatrix.
3. The centralizer (normalizer)
N(SL(n, q)) = kIn | k ∈ F∗q and kn = 1.
4. Z(SL(n, q)) = kIn | k ∈ F∗q and kn = 1.
5. | Z(SL(n, q)) |= gcd(n, q − 1).
6. Let G be a finite group and a ∈ G. Then O(cl(a)) =O(G)
O(N(a)).
7.G
Z(G)∼= I(G).
8. In Sn, the number of distinct cycles of length r isn!
r(n− r)!(r 6 n).
9. Converse of Lagrange’s theorem holds in finite cyclic groups and primepower order.
10. Z(G) =⋂a∈G
N(a).
11. The number of group homomorphism from Zm to Zn is gcd(m,n).
12. Let G be an infinite cyclic group. Then | Aut(G) |= 2.
13. Let G be a finite cyclic group of order n. Then | Aut(G) |= ϕ(n).
14. Aut(Sn) ∼= Sn , Z(Sn) = I for n ≥ 3 and n 6= 6.
15. Aut(Z⊕ Z⊕ · · · ⊕ Z︸ ︷︷ ︸n times
) ∼= GL(n,Z).
3
16. Aut(Zpm ⊕ Zpm ⊕ · · · ⊕ Zpm︸ ︷︷ ︸n times
) ∼= GL(n,Zpm).
17. Aut(Z2 ⊕ Z4) ∼= D8.
18. If G has order n > 1, then |Aut(G)| ≤k∏i=0
(n− 2i) where k = [log2(n−
1)].
19. Let p be a prime number, and let G be a finite abelian group in whichevery element has order p. Then Aut(G) ∼= GL(n,Zp), where n is thedimension of G over Zp.
20. If G is a group of order n and F is any field, then GL(n, F ) contains asubgroup isomorphic to G.
21. Let G = G1 × G2 × · · · × Gn, where the Gi are abelian groups. ThenAut(G) is isomorphic with the group of all invertible n × n matriceswhose (i, j) entries belong to Hom(Gi, Gj), the usual matrix productbegin the group operation.
22. If O(G) = p2q2 and q - p2 − 1, p - q2 − 1, then G is abelian.
23. G is a finite group of order p2q where p and q are distinct primes suchthat p2 1(modq) and q 1(modp). Then G is an abelian group. If pdivides q − 1, then any group of order p2q is abelian.
24. If p does not divide | Aut(G) |, then any group of order pq2 is abelian.
25. If G is a non-abelian finite group, then |Z(G)| ≤ 14|G|.
26. If H and K are subgroups of a finite group G satisfying (|G : H|, |G :K|) = 1, then G = HK.
27. If G is a simple group of order 60, then G is isomorphic to A5.
28. Let G be a group of order pqr, where p > q > r are primes. If p− 1 isnot divisible by q or r and q − 1 is not divisible by r, then G must beabelian (hence cyclic).
29. Abelian groups have exactly one Sylow p-subgroup for each p.
30. The class equation of G is
O(G) = O(Z(G)) +∑
a/∈Z(G)
O(G)
O(N(a)).
4
31. Let G be a non-abelian group of order p3. The number of conjugateclasses of G is p2 + p− 1.
32. Let G be a finite group of order n and p be a prime number such that
p >n
p. Then any subgroup of order p in G is normal in G.
33. Let G be a finite group of order n and p be a prime number such thatp2 does not divide n. Then any subgroup of order p in G is normal inG.
34. The number of non-isomorphic abelian groups of order pn, (p a prime)is p(n) (partition of n).
35. The number of groups of order n is at most nnlog2n.
36. A group of order n can be generated by a set of at most log2n elements.
37. Let n > 1. The content of n is defined to be the product of all distinctprimes dividing n and is denoted by c(n). a ∈ Zn is nilpotent iff c(n)divides a.
38. For any integers a and n with 0 ≤ a < n, then a is idempotent in Zniff a(a− 1) is a multiple of n.
39. Let f(x) = a0 + a1x+ · · ·+ aq−2xq−2 ∈ Fq[x]. Then the number of non-
zero solutions of the polynomial in Fq is equal to q−1−r, r = rank(A)
where A =
a0 a1 a2 · · · aq−2
a1 a2 a3 · · · a0
a2 a3 a4 · · · a1...
......
. . ....
aq−2 a0 a1 · · · aq−1
q−1×q−1
40. Any ring of prime order is commutative.
41. A ring of order p2 ( p a prime) may not be commutative.
42. Smallest non commutative ring is of order 4.
43. Ch(R× S) = l.c.m(ChR,ChS).
44. Z[√−19] is Principle Ideal Doamin but not Euclidean Domain.
45. A field (F,+, . ) is always a Unique Factorization Domain.
5
46. Z[√−5] is an integral domain which is not a Unique Factorization Do-
main.
47. Let F be a field. Then F[x, y] is an example of a Unique FactorizationDomain D which is not a Principle Ideal Doamin.
48. Zn has no non-zero nilpotent iff n is square free.
49. If R has unity 1, then 1 + I is unity ofR
I.
50. Field of reals can be imbedded into the field of complex numbers.
51. If x ∈ [0, 1], then M = f ∈ C[0, 1] | f(x) = 0 is a maximal ideal ofC[0, 1].
52. If F is a field, then the kernel of any evaluation map F [x] −→ F is amaximal ideal.
53. A Euclidean domain possesses unity.
54. Every field is a Euclidean domain.
55. For any commutative ring R with unity,R[x]
〈x〉∼= R.
56. An integral domain R with unity is a field iff R[x] is a Principle IdealDomain.
57. The order of Sylow p-subgroup of Sn is pk where k = [np] + [ n
p2] + · · · .
58. Let σ = (1, 2, 3, · · · , n) ∈ Sn. The number of elements in the conjugateclass of σ is (n− 1)!.
59. If G is a group and A is an abelian group, then there is a bijec-
tion between Hom(G,A) and Hom(
G[G,G]
, A)
where G′ = [G,G] =
〈aba−1b−1〉.
60. S ′n = An for n ≥ 1, A′n = An for n ≥ 5 and A′4 = K4.
61. (GL(n,C))′ = SL(n,C) = (SL(n,C))′.
62. For n ≥ 3, D′2n is 〈s2〉 or 〈s〉 according as n is even or odd.
63. If a normal subgroup N of An (n ≥ 3) contains a 3-cycle, then N = An.
6
64.A4
V4
is cyclic group of order 3.
65.S4
V4
∼= S3.
66.QZ∼= µ∞ = all roots of 1 inC.
67.CR∼= R,
C∗
R∗∼= S1.
68.C∗
S1∼= R∗+.
69.GL(n,C)
SL(n,C)∼= C∗.
70.GL(n,Z)
SL(n,Z)∼= −1, 1.
71.U(n,C)
SU(n,C)∼= S1.
72.O(n,C)
SO(n,C)∼= −1, 1.
73. Let G be an abelian group of order pα.(i) |Aut(G)| = (pα − 1)(pα − p) · · · (pα − pα−1).(ii) The number of subgroups of G of order pβ, 1 ≤ β ≤ α, is
(pα − 1)(pα−1 − 1) · · · (pα−β+1 − 1)
(pβ − 1)(pβ−1 − 1) · · · (p− 1).
74. The number of ring homomorphism from Zn to Zm is 2w(m)−w(
m(n,m)
),
where w(n) denotes the number of prime divisors of n.
75. D2n
〈rk〉∼= D2k.
76. In D2n,
O(Z(D2n)) =
2 if n is even
1 if n is odd.
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77. In D2n,
Number of elements of order 2 =
n+ 1 if n is even
n if n is odd.
78. Number of cyclic subgroup of order n =No.of elements of ordern
ϕ(n).
79. If G is a finite abelian group having a unique subgroup of order p forevery prime divisor p of O(G), then G is cyclic.
80. The group of nth roots of unity is cyclic.
81. A generator for this group is called a primitive n-th root of unity.
82. Let ξ be a primitive nth root of unity, then [Q(ξ);Q] = ϕ(n).
83. The Galois group of the general polynomial of degree n is Sn.
84. Aut(Zn) ∼= U(n).
85. The number of permutation in Sn(n ≥ 4) which commute with σ =(12)(34) is 8(n− 4)!.
86. If σ = (12)(34) in Sn, then | cl(a) |= n(n− 1)(n− 2)(n− 3)
8.
87. Let σ ∈ Sn and let m1,m2, · · · ,ms be distinct integers which appearin the cycle type of σ (including 1-cycles). For each i ∈ 1, 2, · · · , sassume σ has ki cycles of length mi (so that
s∑i=1
kimi = n). The number
of conjugates of σ is
n!
(k1!mk11 )(k2!mk2
2 ) · · · (ks!mkss ).
88. Let σ be the m-cycle (1 2 · · · m). Then σi is also an m-cycle iff (i,m) =1.
89. If G =< x > be a cyclic group of order n. Then a subgroup H is normaliff H = 〈xp〉 for some prime p|n.
90. In D2n, if k|n, 〈rk〉 is normal subgroup in D2n.
91. If |G : H| = m, |G : K| = n. Then lcm(m,n) ≤ |G : H ∩K| ≤ mn.
8
92. If H ≤ K ≤ G, then |G : H| = |G : K||K : H|.
93. If N is a normal subgroup of the finite group G and
(|N |, |G||N |
)= 1,
then N is the unique subgroup of G of order |N |.
94. If G is a simple group of odd order, then G ∼= Zp for some prime p.
95. The finite group G is solvable iff for every divisor n of |G| such that(n,|G|n
)= 1, G has a subgroup of order n.
96. If p is prime, then Sp = 〈σ, τ〉, σ is any transposition and τ is any pcycle.
97. If σ is an m-cycle is Sn. Then |N(σ)| = m(n−m)!.
98. Let p and q be prime numbers with p < q, and let d be the smallestnumber such that pd ≡ 1mod q. Then, any group of order pkq where1 ≤ k < d contains a normal subgroup of order q.
99. If |G| = pdq, then either G contains a normal subgroup of order q or anormal subgroup of order pd.
100. Let G be a group with a subgroup H of index n. If n! < |G|, then thegroup G is not simple.
101. Z(Q8) = 〈−1〉, Inn(Q8) ∼= K4.
102. Z(D8) = 〈r2〉, Inn(D8) ∼= K4.
103. U(2n) ∼= Z2 ⊕ Z2n−2 , n ≥ 3.
104. Let p be a prime and let V be an abelian group with the property thatpv = 0 for all v ∈ V and |V | = pn. Then Aut(V ) ∼= GLn(Fp).
105. For all n 6= 6, we have Aut(Sn) = Inn(Sn) ∼= Sn.
For n = 6, we have |Aut(S6) : Inn(S6)| = 2.
106. Aut(D8) ∼= D8, Aut(Q8) ∼= S4.
107. Sylow p−subgroup of D2n is cyclic and normal for every odd prime p.
108.SL2(F3)
Z(SL2(F3))∼= A4.
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109. If N is a normal subgroup of G, then np(GN
) ≤ np(G).
110. SL2(F4) ∼= A5.
111. np(A×B) = np(A)np(B), where A,B are finite groups.
112. If x is a nilpotent, then 1 + x is unit.
113. Sum of unit and nilpotent is unit.
114. mZ+ nZ = dZ, d = gcd(m,n).
115. mZ ∩ nZ = lZ, l = lcm(m,n).
116. If q ∈ Z be a prime with q ≡ 3(mod 4). ThenZ[i]
〈q〉is a field with q2
elements.
117. Let π be an irreducible element in Z[i]. Then
∣∣∣∣Z[i]
〈πn〉
∣∣∣∣ =
∣∣∣∣Z[i]
〈π〉
∣∣∣∣n .118. If F is a field, then the polynomial ring F [x] is a Euclidean domain.
119. R[x2, x3] is not Euclidean domain.
120. If R is Unique Factorization Domain, then a polynomial ring in anarbitrary number of variables with coefficients in R is also a UniqueFactorization Domain.
121. Let p(x) = a0 + a1x + · · · + anxn ∈ Z[x]. If r
s∈ Q, (r, s) = 1 and
p( rs) = 0, then r|a0 and s|an.
122. If p(x) is a monic polynomial with integer coeffients and p(d) 6= 0 forall integers d dividing the constant term of p(x), then p(x) has no rootsin Q.
123. A polynomial of degree two or three over a field F is irreducible iff ithas no root in F.
124. An irreducible polynomial over Fq of degree n remains irreducible overFqk iff (k, n) = 1.
125. If σ ∈ Sn is the product of disjoint cycles of lenghts n1, n2, · · · , nrwith n1 ≤ n2 ≤ · · · ≤ nr (including its 1-cycles), then the integersn1, n2, · · · , nr are called the cycle type of σ.
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126. Two elements of Sn are conjucate in Sn iff they have the same cycletype.
127. In a integral domain primes implies irreducibility.
128. Field have no primes and irreducible element.
129. Let φ : F → F be a homomorphism of fields. Then φ is either identicaly0 or is injective, so that the image of φ is either 0 or isomorphic to F.
130. If f(x) ∈ F [x] has n distinct roots in its splitting fieldE, thenGal(E/F )is isomorphic to a subgroup of the symmetric group Sn, and so its orderis a divisor of n!.
131. Gal(Zpn|Zp) ∼= Zn.
132. If F is a field and E = F (α), where α is a primitive nth root ofunity, then Gal(E/F ) is isomorphic to a subgroup of U(n) and henceGal(E/F ) is an abelian group.
133. Let F be finite field of order pn. Then the Aut(F ) is cyclic group oforder n.
134. A polynomial f(x) = xn + cn−1xn−1 + · · ·+ c0 is reduced if cn−1 = 0.
If f(x) is a monic polynomial of degree n and if cn−1 6= 0 in F, then its
associated reduced polynomial is f(x) = f(x− cn−1
n).
135. A polynomial f(x) and its associated reduced polynomial f(x) havesame discriminant.
136. The discriminant of a reduced cubic f(x) = x3 + qx+ r is D = −4q3−27r2.
137. If f(x) = x3 + ax2 + bx + c, then its associated reduced polynomial isx3 + qx+ r, where q = b− a2
3and r = 2a3
27− ab
3+ c.
138. The discriminant of f(x) is D = a2b2 − 4b3 − 4a3c− 27c2 + 18abc.
139. Let f(x) ∈ Q[x] be an irreducible cubic with Galois group G anddiscriminant D.
(i) f(x) has exactly one real root iff D < 0, in which case G ∼= S3.
(ii) f(x) has three real roots iff D > 0. In this case, either√D ∈ Q
and G ∼= Z3 or√D /∈ Q and G ∼= S3.
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140. Let n be a positive integer. The sequence of n consecutive compositenumbers is (n+ 1)! + 2, (n+ 1)! + 3, · · · , (n+ 1)! + (n+ 1).
141. The area of the polygon whose vertices are the nth roots of unity in thecomplex plane, when n ≥ 3 is n
2sin2π
n.
142. Let p be a prime and let n be a positive integer, if m ≥ 0 and if∑
isthe sum of the p-adic digits of m, then
nm = n∑
(modp).
143. n3 = n (mod 6) for all n ∈ Z.
144. gcd(a, b) is the smallest positive integer of the form as+ bt.
145. Let n and a be positive integers and let d = gcd(a, n).
Then ax ≡ 1(mod n) has a solution iff d = 1.
146. Let G be a group and let a be an element of G of order n. For eachinteger k between 1 and n, then |ak| = |an−k|.
147. U(2n), (n ≥ 3) is not cyclic.
148. The number of subgroup of the dihedral groupDn (n ≥ 3) is d(n)+σ(n).Where d(n) is the number of positive divisors of n and σ(n) is the sumof the positive divisors of n.
149. Every subgroup of Dn of odd order is cyclic.
150. The group of rational numbers under addition, has no proper subgroupof finite index.
151. Suppose (t, s) = 1. Then U(st) ' U(s)⊕ U(t).
152. U(2n) ' Z2 ⊕ Z2n−2 , n ≥ 3.
153. U(pn) ' Zpn−pn−1 , p odd prime.
154. The centre of a group is characteristic.
155. IfG
Z(G)is cyclic, then G is abelian.
156. If G is non-abelian, then Aut(G) is not cyclic.
157.Z⊕ Z
〈(a, 0)〉 × 〈(0, b)〉' Za ⊕ Zb.
12
158. If A is finite, then
A⊕B ' A⊕ C iff B ⊕ C.
159.D4
Z(D4)' Z2 ⊕ Z2.
160. Let G be the groupQ
Z. Then for every positive integer t, G has a unique
cyclic subgroup of G.
161. If 1 + k is idempotent in Zn, then n− k is an idempotent in Zn.
162. If I is an maximal ideal of a commutative ring R with unity, then I[x]need not be maximal ideal in R[x].
163. If I is a prime ideal of a commutative ring R with unity, then I[x] is aprime ideal of R[x].
164. Let f(x) and g(x) be irreducible polynomial over a field F. If f(x) andg(x) are not associates, then
F [x]
〈f(x) g(x)〉' F [x]
〈f(x)〉⊕ F [x]
〈g(x)〉.
165. In Zn, there are 2d idempotent elements, where d is the number ofdistinct prime divisors of n.
166. Sum of positive integers less than or equal to n and relatively prime to
n isnφ(n)
2.
167. If n = pα11 p
α22 · · · p
αkk , then the sum of all positive divisors of n is
σ(n) =
(pα1+1
1 − 1
p1 − 1
)(pα2+1
2 − 1
p2 − 1
)· · ·(pαk+1k − 1
pk − 1
).
168. If 2 6= d|n, then the number of elements of order d in D2n is φ(d).
169. Number of elements of order d in Zn×Zm is∑
d1|n,d2|mφ(d1)φ(d2), where
d = lcm(d1, d2).
170. If k|n and Uk(n) = x ∈ U(n) | x ≡ 1(mod k), thenU(n)
Uk(n)' U(k).
13
171.Z× Z〈(a, b)〉
is not cyclic if gcd(a, b) 6= 1.
172. Number of elements of order n inQZ
is φ(n).
173. Number of elements of order d in Sn isn!
1α12α2 · · · kαkα1!α2! · · ·αk!,
where lcm(1, 2, · · · , k) = d and αi is the number of i in selected parti-tion.
174. |Aut(D2n)| = nφ(n) where n ≥ 3.
175. Number of conjugate class of D2n is equal ton+6
2if n is even
n+32
if n is odd
176. Zn[i] ' Zn × Zn.
177. If 4|p− 3, then Zp[i] is an integral domain.
178. Sum of two subring need not be subring.
179.Z[i]
〈a+ ib〉' Za2+b2 if gcd(a, b) = 1.
180.Z[i]
〈n〉' Zn[i] ' Z[i]
〈in〉.
181.Z[x]
〈m〉' Zm[x].
182. (GF (pn),+) ' Zp × Zp × · · · × Zp︸ ︷︷ ︸n×times
.
183. If x = a + ib either a = 0 or b = 0 and |x| = p with 4|p − 3, then x isirreducible element of Z[i].
184. If x = a + ib and a 6= 0, b 6= 0 and a2 + b2 = p, then x is irreducible inZ[i].
185. Z[√−d] is not PID if d > 2.
186. Z[√−d] is ED if d = 1, 2.
14
187. [Q(a1/n1
1 , a1/n2
2 , · · · , a1/nkk ) : Q] = lcm(n1, n2, · · · , nk) where ai 6= aj 6=
1.
188. Let A be a group. For every element a ∈ A there exists a unique grouphomomorphism φa : Z→ A that sends 1 7→ a.
189. Let A be a group. Then there exists a bijection between Hom(Z, A)and A. i.e., |Hom(Z, A)| = |A|.
190. For a groupG, the mapHom(G,G1×G2)→ Hom(G,G1)×Hom(G,G2)is a bijection.
191. For a groupG, the mapHom(G1×G2, G)→ Hom(G1, G)×Hom(G2, G)is a bijection.
192. Number of group homomorphism f : Zm → Zn × Zk is gcd(m,n) ×gcd(m, k).
193. Number of group homomorphism f : Zn × Zk → Zm is gcd(m,n) ×gcd(m, k).
194. If n|m, then number of onto homomorphism f : Zm → Zn is φ(n).
195. Number of isomorphism from f : Zn → Zn is φ(n).
196. Number of homomorphism from f : Z→ Zn is n.
197. Number of homomorphism from f : Zn → Z is 1.
198. Number of homomorphism f : Sn → Zm, n ≥ 3 is1 ifm is odd
2 ifm is even.
199. Number of associates of a ∈ Zm are φ(o(a)).
200. List of associates cl(a) = x ∈ Zm | m(x,m)
= o(a).
201. Number of nilpotent elements in Zpn is pn−1.
202. If m = pn11 p
n22 · · · pnrr , then number of nilpotent elements in Zm is
pn1−11 pn2−1
2 · · · pnr−1r .
15
LINEAR ALGEBRA
1. The number of an integer n × n matrix with determinant 1 is cn =1
n+1
(2nn
), n ≥ 1.
2. If sum of the elements in each column (row) of a square matrix of ordern is k, then the sum of the elements in each column (row) of Am willbe km.
3. Let A be an m×n matrix with all entries are distinct. Then the numberof sub matrix of order i× j is
(mi
)×(nj
).
4. The number of sub matrix of all order of a m × n matrix A with all
entries are distinct ism∑i=1
n∑j=1
(mi
)×(nj
)= (2m − 1)(2n − 1).
5. Let A be an m×n matrix with all entries are equal. Then the numberof sub matrix of order i× j is 1.
6. The number of sub matrix of all order of a m × n matrix A with allentries are equal is nm.
7. Let V be a finite-dimensional vector space, and let T : V → V belinear.
(i) If rank(T ) = rank(T 2), then R(T ) ∩N(T ) = 0 and V = R(T )⊕N(T ).
(ii) V = R(T k)⊕N(T k) for some k > 0.
8. Let ∆2n =
a 0 · · · 0 b0 a · · · b 0...
... · · · ......
b 0 · · · 0 a
2n×2n
. Then det(∆2n) = (a2 − b2)n.
9. Let A ∈Mm×n(F ). Then rank(A∗A) = rank(A).
10. If A is an m×n matrix such that rank(A) = n, then A∗A is invertible.
11. For any square matrix A, ‖ A ‖ is finite and, in fact, equals√λ, where
λ is the largest eigenvalue of A∗A.
12. Let A be an invertible matrix. Then ‖ A−1 ‖= 1√λ, where λ is the
smallest eigenvalue of A∗A.
16
13. Let T be a linear operator on V whose characteristic polynomial splits,and let λ1, λ2, · · · , λk be the distinct eigenvalues of T . Then T is diag-onalisable iff rank(T − λiI) = rank(T − λiI)2 for 1 ≤ i ≤ k.
14. A linear operator T on a finite-dimensional vector space V is diagonal-isable iff its Jordan canonical form is a diagonal matrix.
15. The system of linear equations Ax = b has a solution iff b ∈ R(A)(Row space of A).
16. An orthonormal set in an inner product space is linearly independent.
17. Let W1 and W2 be subspaces of a finite dimensional inner product spaceV . Then
(i) (W1 +W2)⊥ = W⊥1 ∩W⊥
2
(ii)(W1 ∩W2)⊥ = W⊥1 +W⊥
2
18. Let A =
[a bc d
]. Then A is diagonalisable iff
(a+ d)2 > 4(ad− bc).
19. For any square matrix A = [aij] of order n,
R(A) = maxRi(A) =n∑j=1
| aij |: 1 ≤ i ≤ n
C(A) = maxCj(A) =n∑i=1
| aij |: 1 ≤ j ≤ n
si = Ri(A)− | aii |
20. For any square matrix A of order n, every eigen value λ of A satisfies| λ− all |≤ sl for some 1 ≤ l ≤ n.
21. For any square matrix A of order n, every eigen value of A satisfies| λ |≤ minR(A), C(A).
22. The eigen value of the tridiagonal matrixA =
a b1 0 · · · · · · 0c1 a b2 · · ·
c2 a b3... · · · · · · a bn−1
· · · cn−1 a
n×n
(n ≥ 3) satisfies the inequality | λ− a |< 2√
maxi| bi | max
i| ci |.
17
23. If A is idempotent. then Rank(A) = Trac(A).
24. If A is symmetric matrix, then Rank(A) is equal to the number ofnon-zero eigen values of A.
25. Test for diagonalization
Let T be a linear operator on an n-dimensional vector space V . ThenT is diagonalizable iff
(i) The characteristic polynomial of T splits.
(ii) For each eigen value λ of T , the multiplicity of λ equals
n− rank(T − λI).
26.
rank
([A BC D
])= rank(A) + rank(D −BA−1C)
= rank(D) + rank(A− CD−1B).
27.
det
([A BC D
])= det(A)det(D −BA−1C)
= det(D)det(A− CD−1B).
28. Let T be a linear operator on a finite dimensional vector space overC. Then T is diagonalisable iff T is annihilated by some polynomial fover C which has distinct roots.
29. A linear operator T on a finite-dimensional vector space V is diagonal-izable iff V is the direct sum of the eigen spaces of T .
30. Any projection E on a vector space V is diagonalizable.
31. If N is nilpotent, then I +N is invertible.
32. If N is nilpotent, then det(I +N) = 1.
33. If A is a matrix and det(I + tA) = 1 for all t, then A is nilpotent.
34. Every singular matrix can be written as a product of nilpotent matrices.
18
35. For an n × n square matrix N with real (or complex) entries, thefollowing are equivalent:
(i) N is nilpotent
(ii) The minimal polynomial of N is λk for some positive integer k ≤ n
(iii) The characteristic polynomial for N is λn
(iv) The only eigen values of N is 0
(v) trace(Nk) = 0 for all k > 0.
36. Dimension of the vector space of n × n matrices when trace is zero isn2 − 1.
37. If V = A = (aij)m×n :n∑j=1
aij = 0, ∀i = 1, 2, · · · ,m
(i) When column is zero then dimension of V is mn−m(ii) If one row and one column is zero,then its dimension is
(m− 1)(n− 1) = mn−m− n+ 1
38. The dimension of symmetric matrices of vector space is∑n =
n(n+ 1)
2
39. The dimension of skew symmetric matrices is∑
(n− 1) =n(n− 1)
2
40. The dimension of symmetric matrices with trace zero isn(n+ 1)
2− 1
41. The dimension of skew-symmetric matrices with trace zero is
n(n− 1)
2
42. Vector space of n × n matrix with complex field over the real numberhave dimension 2n2.
43. Dimension of n × n Hermitian matrices with complex entry over thefield of real numbers is 2n2.
44. Vector space of n×n skew-Hermitian matrices over R, then it’s dimen-sion is n2.
45. Vector space of upper triangular matrices, it’s dimension is∑n
46. Vector space of lower triangular matrices, it’s dimension is∑n
19
47. The dimension of scalar matrices is 1.
48. Dimension of diagonal matrices is n.
49. Dimension of scalar matrices and trace zero is 0.
50. The number of basis in the finite dimensional n vector space over afinite field Zp is,
(pn − 1)(pn − p) · · · (pn − pn−1)
n!
51. The number of subspaces of dimension r is
(pn − 1)(pn − p) · · · (pn − pr−1)
(pr − 1)(pr − p) · · · (pr − pr−1)
52. | adjA |=| A |n−1, if |A| 6= 0
53. | adj. adj · · · adjA |=| A |(n−1)r
54. (i) rank(A) = n ⇒ rank(adjA) = n
(ii) rank(A) = n− 1 ⇒ rank(adjA) = 1
(iii) rank(A) < n− 1 ⇒ rank(adjA) = 0
(iv) rank(AB) ≤ minrank(A), rank(B)
55. Let A and B be any n× n matrix.
Then rank(A)+rank(B)−n ≤ rank(AB) ≤ minrank(A), rank(B).
56. rank(A+B) ≤ rank(A) + rank(B).
57. Let A be an m× n matrix and B be any n× p matrix. Then
Then rank(AB) ≥ rank(A) + rank(B)− n.
58. | A− λI |= λ3 − trace(A)λ2 + trace(adj(A))λ− | A |= 0.
59. Sum of the eigen values = trace(A).
60. Product of eigen values = det(A).
61. If A is nilpotent then A+ λI is always non-singular for any value of λ.
62. The eigen values of symmetric / Hermitian matrix are real.
20
63. The eigen values of skew-Hermitian is purely imaginary.
64. The skew-Hermitian matrices have even rank.
65. The determinant of skew-Hermitian matrix of odd order is zero.
66. For a 2× 2 matrix, the eigen values are
λ =[(a11 + a22)±
√4a12a21 + (a11 − a22)2]
2
67. If T : R2 → R2 defined by
T (x, y) = (ax+ by, cx+ dy) is C-linear iff a = d and b = −c.
68. Let A be an n × n matrix (aij) with complex entries. Suppose |aii| >n∑j=1
|aij|, j 6= i, for all i = 1, 2 · · · , n. Then A is invertible.
69. Let A be an m × n matrix and WA = x ∈ Rm | xA = 0. ThendimWA = m− rank(A).
70. Let A be an m× n matrix and B be an n× p matrix and WAB = x ∈Rm | xAB = 0. Then dimWAB = m− rank(AB).
REAL ANALYSIS
1. Every function is a curve but every curve may not be a function.
2. Card(Z) = Card(N× N) = Card(Q) = ℵ0 (Aleph naught).
3. Card(R) = Card(R× R) = c (Continum).
4. There is a one-one correspondence R→ R× R.
5. There is a one-one correspondence between Rm and Rn for all positiveintegers m and n.
6. Every continuous open mapping of R to R is monotonic.
7. A metric space which is not complete is the space of all rational num-bers, with d(x, y) = |x− y|.
8. Suppose sn is monotonice. Then sn converges iff it is bounded.
9. The product of two convergent series converges, and to the right value,if at least one of the two series coverges absolutely.
21
10. For any set A,
(i) int(A) ⊂ A ⊂ cl(A) = A ∪ A′.
11. (i) If A ⊂ B, then int(A) ⊂ int(B).
(ii) int(A) ∪ int(B) ⊂ int(A ∪B).
(iii) int(A) ∩ int(B) = int(A ∩B).
12. Suppose that zn is a complex sequence with limit z and suppose thatan is a positive sequence such that limn→∞(a1 +a2 + · · ·+an) = +∞.Then lim
n→∞
a1z1 + a2z2 + · · ·+ anzna1 + a2 + · · · an
= z.
13. limn→∞
xn
nk= +∞ if x > 1 and k ∈ N.
14. limn→∞
nkxn = 0 if |x| < 1 and k ∈ N.
15. limn→∞
x1n = 1 if x > 0.
16. For each n ∈ N, limk→∞
kn
2k= 0.
17. For each n ∈ N, limk→∞
2nk
k!= 0.
18. For m ≥ 2,
(i)1
km→ 0 as m→∞.
(ii)1
mk→ 0 as m→∞.
19. Let a1, a2, · · · , ak ∈ R+. Then limn→∞
n√an1 + an2 + · · · ank = maxa1, a2, · · · , ak.
20. Every increasing sequence in Q which is bounded above is a Cauchysequence.
21. Every decreasing sequence in Q which is bounded below is a Cauchysequence.
22. Let∑xn be a series in R such that xn ≥ 0 for all n ∈ N. Then
∑xn
converges iff (sn) is bounded.
23. If (xn) is a decreasing sequence in [0,∞), then∑xn converges iff∑
2nx2n converges.
22
24. π = 12
mint > 0 | eit = 1.
25. Every Boral set is measurable set.
26. There exists measurable set which is not Boral set.
27. For any set A, there exists a measurable set E containing A (i.e., E ⊂A) and such that m ∗ (A) = m(E).
28. Every non-empty open set has positive measure.
29. There exists a non-measurable set.
30. log x =x∫1
dy
y.
31. If f : R→ R is one-one, then d(a, b) = |f(a)− f(b)| is metric.
32. The directional derivative of z = f(x, y) in the direction of unit vectoru = (a, b) is
Duf(x, y) = limh→0
f(x+ ah, y + bh)− f(x, y)
h
33. Duf(x, y) = Of(x, y)·u
34. At a point (x, y), the maximum value of the directional derivativeDuf(x, y) is | Of(x, y) |.
35. If f(x, y) is differentiable, then f has a directional derivative in thedirection of every unit vector u = (a, b) and Duf(x, y) = fx(x, y)a +fy(x, y)b.
36. Let xn be a sequence such that there exist A > 0 and c ∈ (0, 1) forwhich | xn+1 − xn |≤ Acn for any n ≥ 1. Then xn is Cauchy.
37. pn is a Cauchy sequence iff limn→∞
diam EN = 0.
Where EN = pN , pN+1, · · · and diam A = supd(x, y)|x, y ∈ A
38. Let fn is the Fibonacci number sequence, then fn < 2n.
39. Test for continuity at x = 0
Left-hand limit = limh→0
f(0− h)
Right-hand limit = limh→0
f(0 + h)
23
40. Test for differentiability at x = 0
Left-hand derivative = limh→0
f(0− h)− f(0)
−h
Right-hand derivative = limh→0
f(0 + h)− f(0)
h
41. As h→ 0, sin( 1h) oscillates between −1 and 1.
42. As h→ 0, cos( 1h) oscillates between −1 and 1.
43. Let f(x) be a periodic function. If limx→∞
f(x) exists, then f(x) is a
constant.
44. limx→∞
sinx does not exist.
45. If f ′′(x) exists and is continuous in a neighborhood of x = a, then
limh→0
f(a+ h)− 2f(a) + f(a− h)
h2= f ′′(a)
46. If T : Rn −→ Rm is a linear map, then T is differentiable at everya ∈ Rn with T ′(a) = T (a).
47. If f ′′(x) > 0 for all x ∈ R, then f [12(x1 + x2)] ≤ 1
2[f(x1) + f(x2)].
48. Suppose a1 ≥ a2 ≥ · · · ≥ 0. Then the series∞∑n=1
an converges iff the
series∞∑k=0
2ka2k converges.
49. The following functions are uniformly continuous
(i) f : [1,∞) −→ R(ii) Any linear map T : Rm −→ Rn
(iii) f : R −→ R such that f ′(x) exists and is bounded.
50. The Arithmetic-Geometric Mean Inequality for a, b is√ab 6 a+b
2.
51. Let (xn) be a sequence of real numbers and let x ∈ R. If (an) is asequence of positive real numbers with lim
n→∞(an) = 0 and if for some
constant c > 0 and some m ∈ N, we have
| xn − x |6 can, n > m. Then limn→∞
xn = x.
24
52. If 0 < b < 1, then limn→∞
bn = 0.
53. If c > 0, then limn→∞
c1n = 1.
54. limn→∞
n1n = 1.
55. limn→∞
sin n
n= 0.
56. Let (xn) be a sequence of positive real numbers such that l = limn→∞
xn+1
xnexists. If |l| < 1, then (xn) converges and lim
n→∞(xn) = 0
57. xn = 1 + 12
+ 13
+ · · ·+ 1n
for n ∈ N, is a divergent sequence.
58. limn→∞
(1 + 1n)n = e, 2 < e < 3
59. limn→∞
(1 + an)bn = eab.
60. limx→∞
(1 + ax)bx = eab.
61. A continuous periodic function on R is bounded and uniformly contin-uous on R.
62. If f : A −→ R is a Lipschitz function, then f is uniformly continuouson A.
63. If lim|x|→∞
|f(x)| = 0, then f is uniformly continuous.
64. If f has compact support, then f is uniformly continuous.
65. Suppose F is differentiable on I(interval) | f ′(x) |6 M , ∀x ∈ I, thenF is uniformly continuous on I.
66. limx→0
x sin( 1x) = 0.
67. Suppose a ∈ [0, 1], then there exists sequences xn > 0 and yn > 0 suchthat xn → 0, yn → 0 and xynn → a.
68. Suppose that φ : [0,∞)→ R is concave and φ(0) = 0. Then
φ(x+ y) ≤ φ(x) + φ(y), ∀x, y > 0.
25
69. Let f be a continuous real valued function on [0,∞) such that limx→∞
f(x)
exists (finitely). Then f is uniformly continuous.
70. f : R → R continuous function and |f(x) − f(y)| ≥ |x − y| for all xand y. Then f(R) = R.
71. Let f : [0,∞)→ R be a uniformly continuous function with the prop-
erty that limb→∞
b∫0
f(x)dx exists (as a finite limit). Then limx→∞
f(x) = 0.
72. A real valued function of a complex variable either has derivative zeroor the derivative does not exist.
73. Let I ⊆ R be an interval and let f : I → R be monotone on I. Thenthe set of points D ⊆ I at which f is discontinuous is a countable set.
74. If h : R → R satisfies the identity h(x + y) = h(x) + h(y), ∀x, y ∈ Rand if h is continuous at a single point x0, then h is continuous at everypoint of R.
75. If h is a monotone function satisfying h(x + y) = h(x) + h(y), then hmust be continuous on R.
76. If f is bounded and there is a finite set E such that f is continuous atevery point of [a, b] \ E, then f is Riemann integrable.
77. A bounded function f : [a, b] → R is Riemann integrable iff it iscontinuous almost every where on [a, b].
78. The only connected subsets of the real line are the intervals ( open,closed, semi-open or semi-closed ).
79. A non-empty open subset of the complex plane is connected iff anytwo of its points can be joined by polygon lying in the set having itssegments parallel to the coordinates axes.
80. Every compact metric space is complete.
81. A subset S of a metric space X is compact iff S is complete and totallybounded.
82. A metric space is compact iff if every infinite sequence has a limitpoint.
83. Continuous image of connected set is connected.
26
84. Continuous image of a compact set is compact.
85. Suppose E is compact, f is continuous on E iff Graph(f) is compact.
86. Let f be a differentiable real function on (a, b). f is convex if and onlyif f ′ is monotonically increasing.
87. (X, τ) be a compact topological space and F be a closed subset of X.Then F is compact.
88. (X, τ) be a Hausdorff topological space and F be a compact subset ofX. Then F is closed.
89. A complex sequence zn converges iff if it is Cauchy.
90. Suppose limn→∞
fn(x) = f(x) (x ∈ X). Let Mn = supx∈X| fn(x) − f(x) |.
Then fn → f uniformly on X iff if Mn → 0 as n→∞.
91. Suppose fn is a sequence of functions defined on X, and suppose| fn(x) |≤Mn (x ∈ X, n = 1, 2, · · · ). Then
∑fn converges uniformly
on X if∑Mn converges.
92. The limit of uniformly convergent sequence of complex continuous func-tions on a metric space is continuous.
93. f : X → Y continuous. Then the graph of f with the subspace topologyof X × Y is homeomorphic to X.
(i) R is homeomorphic to y = x2. Let f : R→ R , f(x) = x2
94. Let f : X → Y be a homeomorphism and let A ⊂ X. Then f in-duces homeomorphism between A and f(A), (and between X\A andf(X\A)).
(i) [0, 1) is not homeomorphic to (0, 1), take f = I.
95. Any two open intervals of the real line are homeomorphic.
96. Any two closed intervals of the real line are homeomorphic.
97. Let f : X → Y be a continuous bijection. If X is compact and Y isHausdorff then f is a homeomorphism.
98. A bijective continuous map is a homeomorphism iff it is open map.
99. A bijective continuous map is a homeomorphism iff it is closed map.
27
100. A bijective continuous map from a compact metric space to anothermetric space is a closed map and hence is a homeomorphism.
101. If∑cn is convergent and | arg cn |≤ α < π
2, then it converges abso-
lutely.
102. If ϕ : [a, b] −→ R is a step function, then ϕ ∈ R[a, b]
103. If f : [a, b] −→ R is continuous on [a, b], then f ∈ R[a, b]
104. If f : [a, b] −→ R is monotone on [a, b], then f ∈ R[a, b].
105. Two norms are equivalent
(i) ‖ x ‖2≤‖ x ‖1≤√n ‖ x ‖2
(ii) ‖ x ‖∞≤‖ x ‖2≤√n ‖ x ‖∞
(iii) ‖ x ‖∞≤‖ x ‖1≤ n ‖ x ‖∞(iv) 1
d‖ x ‖1≤ 1√
d‖ x ‖2≤‖ x ‖∞≤‖ x ‖2≤‖ x ‖1
106. Box topology on Rw
(i) The box topology is completely regular.
(ii) The box topology is neither compact nor connected.
(iii) The box topology is not first countable.
(iv) Neither is the box topology separable.
(v) The box topology is para compact if the continuum hypothesis istrue.
107. Every metric space is normal.
108. Every compact Hausdorff space is normal.
109. (0, 1N, box topology) is not compact.
110. The orthonormal group
O(n) = A ∈Mat(n;R) : AAT = I is compact.
111. The Special orthogonal group
SO(n) = A ∈ O(n) : det(A) = 1 is a closed subgroup of the compactgroup O(n), and so is itself compact.
112. The Unitary group
U(n) = A ∈Mat(n;C) : T ∗T = I is compact.
28
113. The Special unitary group
SU(n) = A ∈ U(n) : det(A) = 1is a closed subgroup of the compact group U(n), and so is itself com-pact.
114. U(n) =U(1)× SU(n)
< w >, w = e2π/n
115. The Symplectic group
Sp(n) = A ∈Mat(n;H) : T ∗T = I(Here Mat(n;H) denotes the space of n×n matrices with quarternianentries) is compact.
116. A matrix Lie group is any subgroup G of GL(n;C) with the followingproperty; If Am is any sequence of matrices in G and Am converges tosome matrix A, then either A ∈ G or A is not invertible. (i.e, a matrixLie group is a closed subgroup of GL(n;C))
117. The complex orthogonal group
O(n;C) = A ∈ GL(n;C) |< Ax,Ay >=< x, y >, ∀x, y ∈ Cn
.
118. The Special complex orthogonal group
SO(n;C) = A ∈ O(n;C) | det(A) = 1.
119. Let g denote the (n + k) × (n + k) diagonal matrix with ones in thefirst n diagonal entries and minius ones in the last k diagonal entries.
O(n; k) = A(n+k)×(n+k) | AT g A = g
120. A matrix Lie group G is said to be compact if the following two condi-tions are satisfied:
a) If Am is any sequence of matrices in G, and Am converges to a matrixA, then A is in G.
b) There exists a constant c such that for all A ∈ G, | Aij |≤ c for all1 ≤ i, j ≤ n.
29
121.Compact Non− compact
O(n) GL(n;R)
SO(n) GL(n;C)
U(n) SL(n;R)
SU(n) SL(n;C)
Sp(n) O(n;C)
S1 ∼= U(1) SO(n;C)
O(n; k)
SO(n; k)
Sp(n : R)
Sp(n;C)
R,Rn,R∗,C∗
122.Group Connected?
GL(n;C) yes
SL(n;C) yes
GL(n;R) no
SL(n;R) yes
O(n) no
SO(n) yes
SU(n) yes
O(n; 1) no
S0(n; 1) no
123. Let f(x) and g(x) be two differentiable functions with continuous nth
derivatives. Then
dn
dxn(f(x)g(x)) =
n∑k=0
(nk
)dk
dxk(f(x))
dn−k
dxn−k(g(x)).
30
124. U(P, f) =∑n
i=1 Miδi, δi = xi − xi−1.
125. L(P, f) =∑n
i=1miδi, δi = xi − xi−1
126. For every partition P on [a, b], we have L(P, f) 6 U(P, f)
127. m(b− a) 6 L(P, f) 6 U(P, f) 6M(b− a)
where m = inff(x) | x ∈ [a, b]M = supf(x) | x ∈ [a, b]
128.b−∫a
f(x)dx = infU(P, f) : P is partition of [a, b]
129.b∫
a−
f(x)dx = supL(P, f) : P is partition of [a, b]
130. L(P,−f) = −U(P, f)
131. U(P,−f) = −L(P, f)
132.b∫
a−
f(x)dx ≤∫ b−af(x)dx
133. U(P, f) b−∫a
f(x)dx+ ε
134. L(P, f) >b∫
a−
f(x)dx− ε
135. W (P, f) = U(P, f)− L(P, f)
136. If f is Riemann integrable on [a, b], then
b∫a
f(x)dx = limn→∞
n∑r=1
hf(a+ rh), h = b−an
137. If f is integrable on [0, 1], then
1∫0
f(x)dx = limn→∞
1n
n∑r=1
f( rn)
138. L(P, f) 6b∫
a−
f(x)dx 6b−∫a
f(x)dx 6 U(P, f)
31
139. m(b− a) 6b∫a
f(x)dx 6M(b− a), b ≥ a
140. m(b− a) ≥b∫a
f(x)dx ≥M(b− a), b 6 a.
141.b∫a
f(x)dx = µ(b− a) , m ≤ µ ≤M.
142. |b∫a
f(x)dα(x)| ≤ [α(b)− α(a)] maxa≤x≤b
|f(x)|.
143. Let f(x) be a continuous function and α(x) ∈ C1 in a ≤ x ≤ b. Then
b∫a
f(x)dα(x) =
b∫a
f(x)α′(x)dx.
144. A function α(x) is of bounded variation in an interval a ≤ x ≤ b iffα(x) = α1(x)+α2(x), where α1(x) is increasing and α2(x) is decreasingin a ≤ x ≤ b.
145. Let f(x) be an increasing function and α(x) is continuous function on[a, b]. Then
b∫a
f(x)dα(x) +
b∫a
α(x)df(x) = α(b)f(b)− α(a)f(a).
146. Let g(x) be a continuous function and f(x) is an increasing functionon [a, b]. Then
b∫a
f(x)g(x)dx = f(a)
ξ∫a
g(x)dx+ f(b)
b∫ξ
g(x)dx.
147. Let g(x) be a continuous function, f(x) be an increasing function andf(x) ≥ 0 on [a, b]. Then
b∫a
f(x)g(x)dx = f(b)
b∫ξ
g(x)dx, a ≤ ξ ≤ b.
32
148. Suppose sn is monotonic. Then sn converges iff it is bounded.
149. The set of all limit points of a bounded sequence is a compact set(closed set).
150. fn → 0 iff |fn| → 0.
151. fn > 0, ∀n ∈ N, limfn+1
fn= l, l > 0, then lim(fn)
1n = l.
152. If |A| = m, |B| = n, then the number of one-one functions from A toB is n!
(n−m)!(m ≤ n).
153. If f(n) is a positive monotonically decreasing function of n for all n ∈ N,then the two infinite series
∑f(n) and
∑anf(an) converge or diverge
together, a being a positive integer greater than unity.
154. Let f be a function defined for all points in some neighbourhood of apoint b except at the point b itself. Then f(x) → l as x → b iff thelimit of the sequence < f(xn) > exists and is equal to l for any sequence< xn >, xn 6= b for any n ∈ N, converging to b.
155. If f(x) has period T, then af(bx+ c) + d has period T|b| .
156. If f(x) and g(x) has periods T1 and T2, respectively, then af(x)+bg(x)has period T = lcm(T1, T2) provided exists.
COMPLEX ANALYSIS
1. If z = x+ iy, then max|x|, |y| ≤ |z| ≤ |x|+ |y|.
2. Every line or circle in C is the solution set of an equation of the forma|z|2 + wz + wz + b = 0, a, b ∈ R, w ∈ C.
3. If zn is a sequence of complex numbers, then lim zn = w iff lim |zn −w| = 0.
4. Limit point of zeros is an isolated essential singularity of f(z).
5. Limit point of poles is a non-isolated essential singularity of f(z).
6. Poles are isolated of f(z).
7. An entire function f : C −→ C has a removable singularity at z = ∞iff f(z) is constant.
33
8. An entire function f : C −→ C has a pole at z =∞ of order k iff f(z)is a non-constant polynomial of degree k.
9. Let p(z) be a non constant polynomial with real coefficients such thatfor some real number a, p(a) 6= 0 but p′(a) = p′′(a) = 0. Then theequation p(z) has no real root
10. A Mobius transformation which has a unique fixed point in C is calledParabolic.
11. A Mobius transformation has exactly two fixed points, then it is calledloxodromic.
12. The four distinct points z1, z2, z3, z4 in C∞ all lie on a circle or on a lineiff their cross-ratio (z1, z2, z3, z4) is a real number.
13. If f : C −→ C is analytic and f(z) = f(z + z1) = f(z + z2) for allz ∈ C, where z1 and z2 are two non-zero complex numbers such thatz1 or z2 /∈ R, then f is constant.
14. Let f be analytic function on Ω and z0 ∈ Ω such that f ′(z0) 6= 0, thenf is conformal at z0.
15. If f has isolated singular point z0, then
f(z) =∞∑n=0
an(z − z0)n +∞∑n=1
bn(z − z0)n
, 0 < |z − z0| < R
(i) If all b′ns are zero, then the point z0 is said to be removable sigu-larity.
(ii) When an infinite number of the coefficient bn in f(z) are non zero,then z0 is said to be an essential singular point of f .
16. If f has an isolated singularity at z0, then z = z0 is a removable singu-larity iff one of the following conditions holds
(i) f is bounded in a deleted nbh of z0
(ii) limz→z0
(z − z0)f(z) exists
(iii) limz→z0
(z − z0)f(z) = 0.
17. If f has an isolated singularity at z0, then f(z) has an essential singu-larity at z0 iff lim
z→z0f(z) fails to exists either as a finite value or as an
infinite limit.
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18. An entire function f(z) is univalent in C iff f(z) = a0 + a1z(z ∈ C)where a0, a1 are constants with a1 6= 0; that is
Aut(C) = f ∈ H(C) : f(z) = a0 + a1z.
19. If f(z) has a finite limit at z0, then f(z) is a bounded function in someneighborhood of z0.
20. If z0 is pole or essential of f, then z0 is essential of ef .
21. If f(z) is an entire function and have two linearly independent periodsover R, then f(z) is constant.
22. Let f be analytic at a. Then f is one-to-one in some nbh of a ifff ′(a) 6= 0.
23. If f(z) is analytic on a complex domain D, and Re(f ′(z)) > 0 in D,then f is univalent (one to one) in D.
24. Every non-constant entire function omits at most one complex numberas its value, In other words, if an entire function omits two values, thenit is constant.
25. If f is an entire function such that 0 /∈ f(C) and 1 /∈ f(C), then f isconstant.
26. If f(z) is entire, Ref(z) or Imf(z) is bounded on C, then f(z) isconstant.
27. If f(z) is entire, Ref(z) of Imf(z) lies in half-plane, then f(z) is con-stant.
28. Every non constant and non vanishing entire function on C necessarilyhas an isolated essential singularity at ∞.
29. A real valued function of a complex variable either has derivative zeroor the derivative does not exits.
30. f :M→M is analytic such that f(a) = 0 and f(a − 1) = b for some
a ∈ (0, 1) and b ∈M . Then |b| ≤ 1
1 + a− a2.
31. f M→M is analytic, |f(z)| ≤ 1 on |z| = 1, f(a) = 0 and f(−a) = b for
some a ∈ (0, 1) and b ∈ (0, 1]. Then |b| ≤ 2a
1 + a2.
35
32. p(z) is analytic in M with p(0) = 1 and Rep(z) > 0 in M. Then|p′(0)| ≤ 2 and
1− |z|1 + |z|
≤ |p(z)| ≤ 1 + |z|1− |z|
, z ∈M .
33. The complex form of C-R equation fz =∂f
∂z= 0.
34. The complex form of Laplace equation∂2
∂z∂z= 0 and
∂2
∂x2+
∂2
∂y2=
4∂2
∂z∂z.
35. A function which is harmonic and bounded in C must be constant.
36. The range of a non-constant entire function is a dense subset of C.
37. If f has a removable singularity at z0, then Res[f(z) : z0] = 0
38. If f has an isolated singularity at z0 an is f is even in (z − z0), thenRes[f(z); z0] = 0.
39. If a is an isolated singularity of f which is not removable, then a is anessential singularity of ef(z).
40. Let U denote the open unit disc of C, and T its boundary. For any
α ∈ U , define φα(z) =z − α1− αz
. Then
(i) φα is analytic, one to one, onto
(ii) φα(T ) = T
(iii) φ−1α = φ−α
(iv) φ′α(0) = 1− |α|2
(v) φ′α(α) = 11−|α|2
41. Every one-to-one analytic maps f of U onto itself is given by f(z) =λφα(z) for some λ ∈ C with |λ| = 1 and for some α ∈ U .
42. If f : 4 → 4 is analytic and f(a) = 0 for some a ∈ 4. Then
| f(z) |≤| ϕa(z) |, where ϕa(z) =a− z1− az
for z ∈ 4.
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43. Let f : 4 → 4 be analytic having a zero of order n at the origin,f(0) = 0 and | f(z) |< 1 for | z |< 1. Then
(i) | f(z) |≤| z |n
(ii) | fn(0) |≤ n!
and the equality holds in (i) for some point 0 6= z0 ∈ 4 or in (ii) occursiff f(z) = εzn with | ε |= 1.
44. Let f(z) is analytic on the unit disk 4 and satisfies the following twoconditions
(i) | f(z) |≤ 1 for all z ∈ 4(ii) f(a) = b for some a, b ∈ 4.
Then
| f ′(a) |≤ 1− | f(a) |2
1− | a |2
and the equality occurs only if f ∈ Aut(4)
45. sin−1z = −ilog(iz +√
1− z2)
46. cos−1z = −ilog(z + i√
1− z2)
47. tan−1z =i
2log(
i+ z
i− z)
48.d
dzsin−1z =
1√1− z2
49.d
dzcos−1z =
−1√1− z2
50.d
dztan−1z =
1
1 + z2
51. sinh−1z = log(z +√z2 + 1)
52. cosh−1z = log(z +√z2 − 1)
53. tanh−1z =1
2log(
1 + z
1− z)
54.d
dzsinh−1z =
1√1 + z2
37
55.d
dzcosh−1z =
1√z2 − 1
56.d
dztanh−1z =
1
1− z2
GENERAL
1. Any integer n > 1 is either a prime or has a prime factor ≤√n.
2. The number of non-negative integer solutions of the equation ax+by+
cz = n, (a, b, c) = 1, a, b, c 6= 0 is approximatedn2
2abc.
3. HM ≤ GM ≤ AM, 21/a+1/b
≤√ab ≤ a+b
2.
4. For a, b ∈ N, the number of positive integers less than or equal to aand divisible by b is [a
b].
5. L(1) =1
s
6. L(t) =1
s2
7. L(tk) =k!
sk+1, k ≥ 0
8. L(e−at) =1
s+ a
9. L(cos ωt) =s
s2 + ω2
10. L(sin ωt) =ω
s2 + ω2
11. L(e−atcos ωt) =s+ a
(s+ a)2 + ω2
12. L(cos (ωt+ φ)) =s cos φ− ω sin φ
s2 + ω2
13. L(√t) =
√π
2s32
14.df
dt= sF (s)− f(0)
38
15.dkf
dtk= skF (s)− sk−1f(0)− sk−2df
dt(0)− · · · − dk−1
dtk−1f(0)
16.d
dx
β(x)∫α(x)
F (x, t)dt =β(x)∫α(x)
∂
∂xF (x, t)dt+F (x, β(x))
dβ(x)
dx−F (x, α(x))
dα(x)
dx
17. In Fredholm equation. The resolvent kernel is
R(x, t;λ) =∞∑n=1
Kn(x, t)λn−1
where Kn(x, t) =b∫a
K(x, s)Kn−1(s, t)ds , and solution is
y(x) = f(x) + λb∫a
R(x, t;λ)f(t)dt
18. In Volterra equation. The resolvent kernel is
R(x, t;λ) =∞∑n=0
Kn+1(x, t)λn
where Kn(x, t) =x∫t
Kn−1(x, s)K(s, t)ds , and solution is
y(x) = f(x) + λx∫0
R(x, t;λ)f(t)dt
19. Let a = pα11 · · · pαnn and b = pβ11 · · · pβrr .
Then gcd(a, b) =∏p
pmin(αp,βp)
lcm(a, b) =∏p
pmax(αp,βp)
20. lcm(ab, cd) =
lcm(a, c)
gcd(b, d)
21. One-dimensional wave equation
∂2y
∂x2=
1
c2
∂2y
∂t2, −∞ < x <∞, t > 0
y(x, 0) = f(x), yt(x, 0) = g(x), −∞ < x <∞
y(x, t) =1
2[f(x− ct) + f(x+ ct)] +
1
2c
x+ct∫x−ct
g(s)ds
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22. Heat equation
ut = kuxx, −∞ < x <∞, t > 0
u(x, 0) = f(x), −∞ < x <∞
u(x, t) =1
2√πkt
∞∫−∞
f(ξ)e−(x−ξ)2
4kt dξ.
23. If |S| = n. Then there are exactly nn2
number of binary operations onS.
24. The number of commutative binary operations on S is equal to |SX | =nn2+n
2 , where X = (a, a) : a ∈ S.
25. Equation for the angle of the hour handθhr = 1
2(60H +M), where H is the hour and M is the minutes.
26. Equation for the angle of the minute handθmin = 6M.
27. Equation for the angle between the handsδθ =| 1
2(60H − 11M) |