166 |
parametrization of the circle
|z−z0|=r, i.e. the circle centered
at z0 with radius r
|
z(t)=z0+reit,0≤t≤2π
|
166 |
Cr= circle |z−z0|=r
traversed once in the counter-clockwise direction
|
∫Cr(z−z0)ndz={0n≠−12πin=−1
|
186 |
Deformation Invariance theorem.
theorem 8.
|
∫Γ0f(z)dz=∫Γ1f(z)dz
|
187 |
Cauchy's integral theorem.
theorem 9.
f is analytic in a simply connected
domain C and Γ is any loop
(closed contour) in D.
|
∫Γf(z)dz=0
|
|
Cauchy's integral formula.
Holds if f is analytic inside and on the simple
closed positively oriented contour Γ
and z0 is any point inside Γ.
|
f(z0)=12πi∫Γf(z)(z−z0)dz
|
211 |
generalized Cauchy integral formula.
Holds if f is analytic inside and
on the simple closed positively
oriented contour Γ
and z is any point inside Γ.
|
f(n)(z)=n!2πi∫Γf(ζ)(ζ−z)n+1dζ,(n=0,1,2,…)
|
211 |
generalized Cauchy integral formula (equivalent form).
Same conditions except now z0 is any point inside
Γ.
|
f(n)(z0)=n!2πi∫Γf(z)(z−z0)n+1dz,(n=0,1,2,…)
|
211 |
generalized Cauchy integral formula (equivalent form).
Same conditions as previous.
|
2πif(m−1)(z0)(m−1)!=∫Γf(z)(z−z0)mdz,(m=1,2,3,…)
|
215 |
Cauchy estimates for the derivatives of a function f.
Holds if f is analytic inside and on a circle CR of
radius R centered about z0, and
|f(z)|≤M
for all z on CR.
|
|f(n)(z0)|≤n!MRn,(n=1,2,3,…)
|
216 |
mean value property
|
f(z0)=12π∫2π0f(z0+Reit)dt
|