mathj2b

Jordan’s lemma

(i) a function f(z) is analytic at all points z in the upper half plane y≥0 that are exterior to a circle |z| = R_0

(ii) C_r denotes a semicircle z=Re^{i theta}(0<=thetaR_0, where R>R_0

(iii) for all points z on C_r, there is a positive constant M_r such that |f(z)|≤M_r,where:

lim{R right infty}{M_r}=0

Then,for every positive constant a,

lim{R right infty}{int{a}{}{f(z)e^{iaz}dz}}=0

The proof is based on a result that is known as Jordan’s inequality:

int{0}{pi}{e^{-Rsin theta}d theta}<{pi/R}

To verify this inequality, we first note from the graphs of the function y=sin theta and y=2 theta/pi when 0<=theta<=pi/2 that sin theta>=2 theta/pi for all values of theta in that interval. Consequently, if R > 0,

e^{-Rsin theta}

and so

int{0}{pi/2}{e^{-Rsin theta}d theta}<=int{0}{pi/2}{e^{-2R theta/pi}d theta}={pi/{2R}}(1-e^{-R})

Hence

int{0}{pi/2}{e^{-Rsin theta}d theta}<{pi/{2R}} (R>0)