NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India

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1 NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 20

2 Complex Analysis Module: 10: Further theory of analytic functions Lecture: 3: Poisson integral formula for the disc and the plane A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 2 / 20

3 Further theory of analytic functions Poisson Integral Formula A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 3 / 20

4 Poisson Integral Formula We revisit one of the applications of Cauchy integral formula, known as Poisson integral formula from the view point of harmonic functions. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 4 / 20

5 Consequence of Cauchy Integral Formula Poisson Integral Formula Theorem Let f A in z < ρ and z = re iθ in a domain D that contains z < ρ. Then f (re iθ ) = 1 2π (R 2 r 2 )f (Re iφ ) 2π 0 R 2 2πR cos(θ φ) + r 2 dφ, where 0 < R < ρ. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 20

6 Poisson Integral Formula Proof Let D ρ : z : z < ρ. Let C be a circle of radius R lying inside D ρ. Let z be a point in the interior of C. Then by Cauchy Integral Formula f (z) = 1 2πi C f (w) w z dw A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 6 / 20

7 Poisson Integral Formula Proof... Let z be the inverse of z with respect to the circle C. Hence z z = R 2 or z = R2 z Choosing the point w = Re iφ gives Clearly z lies outside C. z = w 2 z = ww z Hence by Cauchy Goursat Theorem 0 = 1 f (w) 2πi w z dw C A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 7 / 20

8 Poisson Integral Formula Proof... From the above two integrals, f (z) = 1 f (w) 2πi C [ ] 1 w z 1 w z dw. Writing z = re iθ, w = Re iφ and dw = iwdφ gives the required answer. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 8 / 20

9 Poisson Integral Formula for Harmonic functions Let U be a real valued function defined on the circle z = R and continuous there except for a finite number of jump discontinuities. Then the function u(re iθ ) = 1 2π 2π 0 U(Re it )(R 2 r 2 ) R 2 2rR cos(t θ) + r 2 dt is harmonic inside the circle, and as re iθ approaches any point on the circle where U is continuous, u(re iθ ) approaches U at that point. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 9 / 20

10 Poisson Integral Formula for Harmonic functions The part of the integrand P(R, r, t θ) = is called Poisson Kernel of the integral. This has the property 1 2π P(R, r, t θ)dt = 1. 2π 0 Hence Poisson Integral Formula is written as R 2 r 2 R 2 2rR cos(t θ) + r 2 u(re iθ ) = 1 2π U(Re it )P(R, r, t θ)dt. 2π 0 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 20

11 Poisson Integral Formula for Harmonic functions This Poisson Integral Formula is obtained for the circular disc. Note that this Poisson Integral Formula for Harmonic functions can be obtained from the corresponding formula for the analytic functions by comparing the real parts. We now provide the series representation for the Poisson Integral Formula. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 11 / 20

12 Series representation for Poisson Kernel From the Poisson Kernel, we have P(R, r, t θ) = R 2 r 2 R 2 2rR cos(t θ) + r 2 = Re w + z w z. Since we get w + z (1 w z = + z ) ( 1 + z ) 1 ( = 1 + z w w w w + z w z = ( z w n=1 ) n. ) ( z ) n w n=0 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 12 / 20

13 Series representation for Poisson Kernel Hence ( ) Re w + z ( z ) n w z = Re w n=1 ( ) ( r ) n = Re e nθ nt R = where w = re iθ and z = Re it. n=1 ( r ) n cos(nθ nt), R n=1 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 13 / 20

14 Series representation for Poisson Integral Formula This gives U(r, θ) = 1 2π 2π 0 ( ) ( r ) n cos(nθ nt) U(R, t)dt R n=1 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 14 / 20

15 Series representation for Poisson Integral Formula This can be rewritten as where U(r, θ) = a 0 + ( r ) n (an cos nθ + b n sin nθ), R n=1 a 0 = 1 2π 2π a n = 1 2π b n = 1 2π 0 2π 0 2π 0 u(r, t)dt u(r, t) cos nt dt u(r, t) sin nt dt. Note that this representation is same as the Fourier representation. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 15 / 20

16 Poisson Integral Formula for Half Plane We are also interested in Poisson Integral Formula for Half Plane. Before providing the Poisson Integral Formula for the half plane, we provide the corresponding Cauchy integral formula. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 16 / 20

17 Cauchy Integral Formula for Half Plane Let f be analytic in the upper half plane H := {z : Im z > 0}. Let α and M be constants such that f satisfies z α f (z) < M. Let C R be a positively oriented circle of Radius R on H centered at origin. By Cauchy Integral Formula f (z) = 1 f (s)ds 2πi C R s z + 1 R f (t)dt 2πi R t z. Since f (z) < M/R α, as R, the first integral approaches zero. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 17 / 20

18 Cauchy Integral Formula for Half Plane This gives the Cauchy Integral Formula for the Half Plane as f (z) = 1 2πi f (t)dt, Im z > 0. t z When z is below real axis, the right hand side of above formula is zero. Hence, when z is above real axis, for some arbitrary complex number k, we get f (z) = 1 ( 1 2πi t z + k ) f (t)dt, Im z > 0. t z A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 18 / 20

19 Poisson Integral Formula for Half Plane Now we consider the case k = 1. This gives f (z) = 1 π yf (t) dt, y > 0. t z 2 Writing f (z) = u(x, y) + iv(x, y), we get by comparing the real parts, u(x, y) = 1 yu(t, 0) π t z 2 dt = 1 yu(t, 0) π (t x) 2 dt, y > 0. + y 2 This formula is known as Poisson Integral Formula for the half plane. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 19 / 20

20 Schwarz Integral Formula for Half Plane Now we consider the case k = 1. This gives f (z) = 1 (t x)f (t) iπ t z 2 dt, y > 0. Writing f (z) = u(x, y) + iv(x, y), we get by comparing the imaginary parts, v(x, y) = 1 π (x t)u(t, 0) (t x) 2 dt, y > 0. + y 2 This formula is known as Schwarz Integral Formula for the half plane. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 20 / 20