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Give the power series expansion of f(z) = z^2 * exp(z-1) about
1 and find its radius of convergence.

f(z) = +(n=0 to n=oo) a[n] * (z-1)^n

a[n] = 1/(n-2)! + 2/(n-1)! + 1/(n!)

hint: z^2 * exp(z-1) = ((z-1) + 1)^2 * exp(z-1)

a[n] = 1/(n-2)! + 2/(n-1)! + 1/(n!)

hint: z^2 * exp(z-1) = ((z-1) + 1)^2 * exp(z-1)

f(z) = 6*sin(z^3) + z^3*(z^6 - 6)

Find the multiplicity of 0.

Find the multiplicity of 0.

15

Determine the multiplicity of all zeros of cos.

1

hint: cos(z) = -sin(z - pi/2)

hint: cos(z) = -sin(z - pi/2)

Determine the multiplicity of all zeros of sin.

1

page 87 in OLDTIMER

page 87 in OLDTIMER

Give the power series expansion of cosine about pi/2.

+(k=0 to k=oo) [ (-1)^(k+1) * (z-pi/2)^(2k+1) / (2k+1)! ]

hint: cos(z) = -sin(z - pi/2)

hint: cos(z) = -sin(z - pi/2)

Let G be an open connected subset of |C.

Let f,g : G -> |C be diffable.

What theorem can be used to prove that:

f*g=0 on G ==> f=0 on G or g=0 on G

Let f,g : G -> |C be diffable.

What theorem can be used to prove that:

f*g=0 on G ==> f=0 on G or g=0 on G

Let G be an open connected subset of |C.

Let f : G -> |C be diffable.

(1) {z:-G : f(z)=0} has a limit point in G

(2) /\z:-G f(z)=0

thesis: (1)=>(2)

REMARK: MRW hasn't done this proof yet. 2001.02.23; 2001.05.15

Let f : G -> |C be diffable.

(1) {z:-G : f(z)=0} has a limit point in G

(2) /\z:-G f(z)=0

thesis: (1)=>(2)

REMARK: MRW hasn't done this proof yet. 2001.02.23; 2001.05.15

Let W be a non-empty collection of sets such that

(1) A,B:-W => AnB:-W

(2) A,B:-W => A\B is a finite disjoint union of sets in W

Let y : W -> |R* be additive.

Can we conclude that y is finitely additive?

(1) A,B:-W => AnB:-W

(2) A,B:-W => A\B is a finite disjoint union of sets in W

Let y : W -> |R* be additive.

Can we conclude that y is finitely additive?

NO

W = { O, {1}, {2}, {3}, {1,2,3} }

y({1,2,3}) = 1

y(O) = y({1}) = y({2}) = y({3}) = 0

y is additive but not finitely additive

W = { O, {1}, {2}, {3}, {1,2,3} }

y({1,2,3}) = 1

y(O) = y({1}) = y({2}) = y({3}) = 0

y is additive but not finitely additive

G c |C; f : G -> |C; a:-|C

What does it mean that function f has an isolated singularity at a?

What does it mean that function f has an isolated singularity at a?

(1) a doesn't belong to G

(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B

(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B

G c |C; f : G -> |C; a:-|C

(1) a doesn't belong to G

(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B

What is the verbal description of this situation?

(1) a doesn't belong to G

(2) there exists B = {z:-|C : 0 < |z-a| < r} c G such that f is diffable on B

What is the verbal description of this situation?

function f has an isolated singularity at a

Let G be a subset of |C. Let f : G -> |C. Let a:-|C.

What does it mean that function f has a removable isolated singularity at a?

What does it mean that function f has a removable isolated singularity at a?

There exists a number r > 0 and function g : B(a;r) -> |C such
that:

(1) a doesn't belong to G

(2) B(a;r) \ {a} c G

(3) g is diffable

(4) g=f on B(a;r)\{a}

(1) a doesn't belong to G

(2) B(a;r) \ {a} c G

(3) g is diffable

(4) g=f on B(a;r)\{a}