deck 1 Flashcards

1
Q

f is continuous at z if and only if ?

A

limf(z) = f(z_o)

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2
Q

f is differentiable at x if and only if ?

A

-

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3
Q

state the Cauchy equations

A

-

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4
Q

Cauchy thm. f being differentiable implies two things

A

partial derivertives exist and cauchy eqns hold

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5
Q

The converse thm ?

A

partial derivertives exist and continuous and cauchy eqns hold THEN f is differentiable

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6
Q

what is the ratio test of a power series?

A

-

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7
Q

what is the root test of a power series?

A

-

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8
Q

what is the sum formula of exp(x)

A

Σ (z^n/n!)

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9
Q

what is the sum forumla for cos(z)?

A

-

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10
Q

what is the sum formula for sin(z)?

A

-

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11
Q

log(z) = ?

A

ln|z| + iarg(z)

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12
Q

Log(z) =?

A

ln|z| + iArg(z)

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13
Q

cos(A+B) = ?

A

cosAcosB - sinAsinB

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14
Q

sin(A+B) = ?

A

cosAsinB + cosBsinA

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15
Q

what is the contour integration formula?

A

-

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16
Q

contour is closed if ?

A

γ(a)=γ(b)

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17
Q

lengh(γ) = ?

A

-

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18
Q

what is the fundamental theorem of contour integration?

A

-

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19
Q

what is the winding number? which rotation is negative ?

A

-

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20
Q

ω(r,0) = ?

21
Q

ω(r,z_o) = ?

22
Q

what is the estimation lemma?

23
Q

what is Cauchey’s theorem? (not generalised)

24
Q

what is Cauchey’s generalised theorem ?

25
Q

what is Cauchey’s integral formula for a circle?

26
Q

what is Taylor’s theorem

27
Q

what is caucheys estimate

28
Q

what is liouvilles theorem ?

29
Q

what is the fundamental theorem of algebra

30
Q

what is laurents theorem ?

31
Q

what is a singularity?

32
Q

what is a pole and its order?

33
Q

if f(z) = p(z)/q(z) when i and ii hold, then f has a pole of order m at z_o. what are the conditions i, ii ?

34
Q

Res(f,z_o) = ?

35
Q

Res(f,z_o) also equals ? hint answer is an integral

36
Q

what is caucheys residue theorem?

37
Q

if f(z) = p(z)/q(z), and more conditions, what are they? then Res(f,z_o) = ?

38
Q

dz = ?

39
Q

if the integrals of z_o to z_1 are different on different paths, what does this imply?

40
Q

what is the sum formula of 1/(1-z)

41
Q

outline the comparison test

42
Q

b^a = ?

A

exp(aln(b))

43
Q

a smooth path is ?

A

differentiable

44
Q

what does Res(f,z_o) also equal ? hint answer is a limit

45
Q

define a singularity

46
Q

define the principle part of the laurent series

47
Q

what is the integral from -∞ to ∞ f(x)dx defined to be ? hint its a limit.
and what is the principle value ?

48
Q

what is the formula that ensures the principle value is equal to the defined value?