deck 1 Flashcards
f is continuous at z if and only if ?
limf(z) = f(z_o)
f is differentiable at x if and only if ?
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state the Cauchy equations
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Cauchy thm. f being differentiable implies two things
partial derivertives exist and cauchy eqns hold
The converse thm ?
partial derivertives exist and continuous and cauchy eqns hold THEN f is differentiable
what is the ratio test of a power series?
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what is the root test of a power series?
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what is the sum formula of exp(x)
Σ (z^n/n!)
what is the sum forumla for cos(z)?
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what is the sum formula for sin(z)?
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log(z) = ?
ln|z| + iarg(z)
Log(z) =?
ln|z| + iArg(z)
cos(A+B) = ?
cosAcosB - sinAsinB
sin(A+B) = ?
cosAsinB + cosBsinA
what is the contour integration formula?
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contour is closed if ?
γ(a)=γ(b)
lengh(γ) = ?
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what is the fundamental theorem of contour integration?
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what is the winding number? which rotation is negative ?
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ω(r,0) = ?
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ω(r,z_o) = ?
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what is the estimation lemma?
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what is Cauchey’s theorem? (not generalised)
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what is Cauchey’s generalised theorem ?
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what is Cauchey’s integral formula for a circle?
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what is Taylor’s theorem
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what is caucheys estimate
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what is liouvilles theorem ?
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what is the fundamental theorem of algebra
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what is laurents theorem ?
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what is a singularity?
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what is a pole and its order?
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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 ?
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Res(f,z_o) = ?
b_1
Res(f,z_o) also equals ? hint answer is an integral
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what is caucheys residue theorem?
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if f(z) = p(z)/q(z), and more conditions, what are they? then Res(f,z_o) = ?
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dz = ?
iexp(it)
if the integrals of z_o to z_1 are different on different paths, what does this imply?
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what is the sum formula of 1/(1-z)
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outline the comparison test
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b^a = ?
exp(aln(b))
a smooth path is ?
differentiable
what does Res(f,z_o) also equal ? hint answer is a limit
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define a singularity
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define the principle part of the laurent series
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what is the integral from -∞ to ∞ f(x)dx defined to be ? hint its a limit.
and what is the principle value ?
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what is the formula that ensures the principle value is equal to the defined value?
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