Review of Mathematical Physics II
Review of Mathematical Physics II 1Contents INote: MatricesBasic
OperationsSpecial types of MatricesCoordinate
transformationEigenvalues Problem2Contents IISeries Convergent
{un}Tests for absolute convergent series {s||n}Leibniz criterion
for S(-1)nan.Uniformly convergent {un(z)}Bernoulli function &
Euler-Maclaurin FormulaAsymptotic Series {sn(x)}3Contents
IIIFunctions of Complex variables:Complex
AlgebraDifferentiableAnalyticCauchys integral theorem &
formulaMoreras theoremTaylor & Laurent expansionBranch point,
pole and essential singularityConformal mapping &
mappingResidue Theorem4MatricesBasic Operations2
matrices:A+B,A-B,A*B1 matrix:A, AT, tr(A), |A|, A-1.How about A=BC
for these operators? E.g. tr(BC)=tr(CB). element E.g.
(AT)ij=Aji.5MatricesSpecial types of MatricesNormal Matrices:
NN=NN.Properties: have complete orthogonal eigenvectors.f(N)=S
f(li)|yi >a=R(a)|u>a , 6
SeriesConvergent {un}:If for every e >0 there is a positive
number N such that |un-u|N.Cauchys criterionTests for absolute
convergent series {s||n}Properties of Conditionally and Absolutely
convergent series.s||n s|| , we can get sns.Sa+bnSa+Sb.Sun=Sup(n)
for absolutely convergent series.SSunvm=Scl if one of them is
absolutely convergent series.Leibniz criterion for S(-1)nan.
7SeriesUniformly convergent {un(z)}If for every e >0 there is
a positive integer N such that |fn(z)-f(z)|N and every point z in
D.Can be tested by Weierstress testAbel test.Bernoulli function
& Euler-Maclaurin Formula ()Asymptotic Series {sn(x)}xnxnRn(x)0
x n Rn(x)
8Functions of Complex variablesComplex algebra:Representations
of z: z=x+iy f(z)=u+iv.z=|z|eif.Complex algebra:1 variable: z*,
arg(z), z-1.2 variables: z1+z2, z1z2.9Functions of Complex
variablesComplex algebra10DifferentiableAnalyticCauchy-Riemann
condition.fx & fy are continuous.Cauchys integral
theoremCauchys integral Formula
Moreras theorem.Laurent expansionTaylor expansionResidue
TheoremProduct theorem: |AB|=|A||B|11
Tr(AB)=Tr(BA), AB=(BA),A-1B-1=(BA)-112
Normal matrix has complete orthogonal eigenvectors13
A matrix has complete orthogonal eigenvectors is a Normal
matrix14
Leibniz criterionIf an>0, an is monotonically decreasing (for
sufficiently large n) and limnan=0, then the series S1(-1)n+1an
converges.
15
Weierstrass M test ()If we can construct a (i)convergent series
of numbers S1Mi, in which (ii) Mi|ui(z)| for all z in a closed
region D, our series S1ui(z) will be absolutely and uniformly
convergent in D.Proof:
1617
Abels testLet {un(x)} be a sequence of functions. If un(x) can
be written as un(x) =anfn(x),San is convergent. ( an0), fn(x) is a
monotonic decreasing sequence, fn(x) is bounded in some region
[a,b] (i.e. 0fn(x) M).Proof:18
1. Weierstrass 2. bnfn(x)3. Remainder of Taylor Expansion (real
number)19
Proof: ( mathematical induction) n=1 LHS:f(x)-f(a),
RHS:R1(x)=f(x)-f(a): n=mF(x)=f(1)(x)n=m+1f(x)n=1(2)nUniform
convergence (fn(z)f(z) ) If fn(z)f(z) in zD, c fn(z)dz c f(z)dz. If
fn(z)f(z) and f n(z)g(z) in zDf n(z)f (z) If fn(z)f(z) and fn(z) is
continuous in zDf(z) is continuous20Uniform convergence (3)21
Uniform Convergence (1)22
Uniform convergence (2)23
Cauchy-Riemann Condition24
z0For eq. (1)Cauchys Integral Theorem25
Cauchys integral formula26
General Cauchys integral FormulaMathematical Induction27
Moreras Theorem28
Taylors expansionuniqueness 29
30
Laurent ExpansionTaylors expansion z0{an}an f(n)/n!contourr
