Notes 8 Analytic Continuation - Courses.egr.uh.edu
ECE 6382y 1 21xFall 2020David R. JacksonNotes 8Analytic ContinuationNotes are adapted from D. R. Wilton, Dept. of ECE1
Analytic Continuation of Functions We define analytic continuation as the process of continuing afunction off of the real axis and into the complex plane such that theresulting function is analytic. More generally, analytic continuation extends the representation of afunction in one region of the complex plane into another region,where the original representation may not have been valid.For example, consider the Bessel function Jn (x). How do we define Jn (z) so that it is computable in some regionand agrees with Jn (x) when z is real?2
Analytic Continuation of Functions (cont.) One approach to extend the domain of a function is to use Taylor series. We start with a Taylor series that is valid in some region. We extend this to a Taylor series that is valid in another region.Note:This may not be the easiest way in practice, but it always works in theory.3
Analytic Continuation of Functions (cont.) 1f ( z ) zn ,1 zn 0valid there, find coefficients of a new series :f ( z) bm ( z )1 m2m 0by usingyz 1 Expand about z 12 . Since both series are Original geometric seriesExampletwo alternativerepresentations,z 12 Rc 32 1 2321xuse the above seriesrepresentation n 1 dm1 1 dm fn m bm znnnnmz121 ()()() mm! dz m z mm!!1dz10 nnm z 1z 2 1 n!1 n m ( ) m! n m ( n m ) ! 2 22dm n(Note m z 0 , m n: n 0 ,1, ,m 1 )dzThe coefficients of the new series --- with extended region of convergence --- are determinedfrom the coefficients of the original series, even though that series did not converge in theextended region. The information to extend the convergence region is contained inthe coefficients of the original series --- even if it was divergent in the new region!4
Analytic Continuation of Functions (cont.)Example (cont.)Another way to get the Taylor series expansion:f ( z) bm ( z )1 m2m 0,z 12 1 dm 1 bm m! dz m 1 z z 1/ 232so thatbm11 2 (1)( 2 )( 3) ( m ) m 1m! 3 (1 z )z 1/ 2 f ( z) 2 3 m 0m 1( z 12 )m,z 12 32m 1Note:This is sort of “cheating” in thesense that we assume wealready know a closed formexpression for the function.5
Analytic Continuation of Functions (cont.)Example (cont.) f ( z ) znn 0Here we show the continuationof f ( z ) from its power seriesyrepresentation in the regionz 1 into the entire complexplane using Taylor series. 1 21 If the singularities are isolated, we can continue any functioninto the entire complex plane via a sequence of continuationsusing Taylor and / or Laurent series !x6
The Zeros of an Analytic Function are Isolated(The zeros cannot be arbitrarily close together.)Proof of theorem : Assume that f ( z ) is analytic in a connected region A , and suppose f ( z0 ) 0.Then f ( z ) has a Taylor series f ( z0 ) a1 ( z z0 ) a2 ( z z0 ) 2 with a0 f ( z0 ) 0 at z0 . More generally, we may have a zero with multiplicity N such that f ( z ) hasa Taylor series as :f ( z ) aN ( z z0 ) N aN 1 ( z z0 ) N 1 1( a N f ( N ) ( z0 ) 0 )N!( z z0 ) N g ( z ), g ( z0 ) 0, and g ( z ) is analytic in A (it is represented by a ( z z0 ) N aN aN 1 ( z z0 )1 converging Taylor series).Since g ( z ) is analytic it is also continuous. Since g ( z0 ) 0 , g ( z ) cannot vanish within asufficiently small neighborhood of z0 . That is, the zeros of f ( z ) must be isolated .The only exception is if the function f (z) is identically zero.7
The Zeros of an Analytic Function are Isolated Example :f ( z ) sin (1 z ) has zeros The function at z1, n 1, 2, nπThis function cannot be analytic at z 0 since the zeros accumulate there andhence are not isolated there.y1/ πxThe origin is an “accumulation point” for the zeros.8
Analytic Continuation PrincipleTheorem of analytic continuation:Assume that f(z) and g(z) are analytic in a connected region A, and f (zn ) g(zn)on a set of points zn in A that converge to a point z0 in A.In other words, there can be only one functionthat is analytic in A and has a defined set ofvalues at the converging points zn.Then g(z) f (z) in A.Proof:z0znAg(z)Note: Defining zn on a contour in A also suffices. Construct the difference function g(z)-f(z),which in analytic in A. This function musthave a Taylor series at z0. This function has all zero coefficients andthus be zero; otherwise, the analyticdifference function must have isolatedzeros because it is analytic – which itdoes not, by assumption. By continuing the Taylor series that haszero coefficients (analytic continuation),the difference function must be zerothroughout A.9
Analytic Continuation Principle (cont.)Corollary (extending a domain from A to A B)Assume that f (z) is analytic in A and g(z) is analytic in B, and the twodomains overlap in a region A B, and f(z) g(z) in A B. f ( z ) , z ADefine h ( z ) g ( z ) , z BThen h(z) is the only analytic function in A B that equals f (z) on A.f ( z) g ( z)Af ( z)A BA Bg ( z)The function h(z) uniquelyextends the domain of f (z)from A to A B.B10
Analytic Continuation Principle (cont.)Then h(z) is the only analytic function in A B that equals f (z) on A.Proof: The function h is analytic in the region A B and also equals f(zn) on any set ofconverging points in the intersection region. The theorem of analytic continuation thus ensures that h is unique in A B.Converging pointsA BAf ( z)A Bg ( z)B11
Analytic Continuation Principle (cont.)Example (example of theorem)The function sin(x) is continued off the real axis.yeiz e iz g ( z ) sin ( z ) 2ixLine segmenteix e ix f ( x ) sin ( x)2iThe function g(z) is the only one that is analytic in the blue region of thecomplex plane and agrees with sin(x) on any segment of the real axis.12
Analytic Continuation Principle (cont.)Example (example of theorem)The Bessel function Jn(x) is continued off the real axis.kn 2k 1)( z g ( z ) J n ( z) knk!! () 2 k 0 yxLine segmentkn 2k 1)( x f ( x ) J n ( x) k 0 k!( n k ) ! 2 The function g(z) is the only one that is analytic in the blue region of thecomplex plane and agrees with Jn (x) on any segment of the real axis.13
Analytic Continuation Principle (cont.)Example (of corollary)The function Ln(z) (principal branch) is continued beyond a branch cut.f ( z ) Ln ( z ) ln ( r ) iθ ,y π θ πNote: “Ln” denotes the principal branch.Region of overlap(second quadrant)Branch cut for fxg ( z ) ln ( z ) ln ( r ) iθ π / 2 θ 3π / 2Branch cut for gIn the blue region (second and third quadrants), g(z) is analytic. Also, g(z) agrees withthe function f (z) in the second quadrant. The original function f (z) is not analytic inthe entire blue region.14
Analytic Continuation Principle (cont.)Example (cont.)The function g(z) is the only function that is analytic in the entire left-halfplane and agrees with Ln(z) in the second quadrant.yxg ( z ) ln ( z ) ln ( r ) iθ , π / 2 θ 3π / 215
Analytic Continuation Principle (cont.)Example (of corollary)The function Yn (z) (Bessel function of the second kind) is continued beyond abranch cut.yBranch cutxLn ( z ) ln ( r ) iθ π θ πYn ( z ) z 1 n 1 ( n k 1) ! z J ( z ) Ln γ π nk! 2 2 π k 0212 k n1 z ( 1) Φ ( k ) Φ ( n k ) π k 0k !( n k )! 2 k1 11Φ ( p ) 1 2 3p2 k nγ 0.577216(Euler’s constant)( p 0)16
Analytic Continuation Principle (cont.)Example (cont.)The “extended Bessel function” is analytic within the blue region.yNote: In the third quadrant, this “extended”function will not be the same as the usualBessel function there.In the third quadrant :xYnext ( z ) Yn ( z ) 2π z 1 n 1 ( n k 1) ! z extYn ( z ) J n ( z ) ln γ k!π 2 2 π k 021k π / 2 θ 3π / 22k n1 z ( 1) Φ ( k ) Φ ( n k ) k !( n k ) ! 2 π k 0 ln ( z ) ln ( r ) iθJ n ( z )( 2π i )2k n17
Analytic Continuation Principle (cont.)ExampleHow do we extend F(x) to arbitrary z? F ( x ) e xt J 0 ( t ) dt, x 00Note: The integral does not converge for x 0.yOriginal domain: x 0xIdentity : xt e J 0 ( t ) dt012x 1, x 018
Analytic Continuation Principle (cont.)Example (cont.)Here we define:F ( z) (z12) 11/ 2This is defined everywhere in the complex plane (except on the branch cuts).yi(x2) 11/ 2 x2 1x iNote:The shape of the branch cutsis arbitrary here.19
Analytic Continuation Principle (cont.)Example (cont.)Here we define:F ( z) 12z 1Note: Re z 2 1 0This corresponds to using vertical branch cuts.With these branch cuts,()Re z 2 11/ 2y 0The derivation (omitted) is similar tothat of the Sommerfeld branch cuts.i(x2) 11/ 2 x2 1x(2) z 11/ 2 z2 1 i(with these branch cuts)20
Analytic Continuation Principle (cont.)ExampleHow do we extend F(x) to arbitrary z?x1F ( x ) dtt1Im ( t ) (real valued for real x 0)Original domain: x 01xRe ( t )F ( x ) ln ( x ) ln (1) Ln ( x ) Ln (1)Recall : ln ( z ) Ln ( z ) i ( 2π n )Note : Ln ( z ) means π arg ( z ) πF ( x ) Ln ( x )21
Analytic Continuation Principle (cont.)Example (cont.)Analytic continuation:F ( z ) Ln ( z )(This agrees with F(x) on the real axis.)yxBranch cut22
Analytic Continuation Principle (cont.)Example (cont.)zFrom another point of view:1F ( z ) dtt1Im ( t )Cz zzz1Re ( t )zWe analytically continue F(z) from the real axis.We require that the path is varied continuously as z leaves the real axis.23
Analytic Continuation Principle (cont.)Example (cont.)zIm ( t )CF z F ( z) zz1C ( z)z 1F ( z ) dtt1Re ( t )zAs we encircle the pole at the origin, we get a different result for the function F(z).1 z dz C1 z dz C1 dzFzF ()( z ) 2π i zCThis is why we need a branch cut in the z plane, with a branch point at z 0.24
Analytic Continuation Principle (cont.)ExampleHow do we interpret cosh-1(z) for arbitrary z?())( (cosh 1 ( x ) Ln x x 2 1 , x 1(Note : cosh 1 (1) 0) )Note : Ln x x 2 1 is real arg x x 2 1 0yOriginal domain: x 11xxRecall : Ln ( z ) means π arg ( z ) π25
Analytic Continuation Principle (cont.)Example (cont.)Analytic continuation:(cosh 1 ( z ) Ln z z 2 1)yz1xx(cosh 1 ( x ) Ln x x 2 1)26
Analytic Continuation Principle (cont.)Example (cont.)()z2 11/ 2(yWhere would branch cuts be?cosh 1 ( z ) Ln z z 2 1 z2 1Sommerfeld branch cuts1 1Note:The function cosh-1(z) is discontinuousas we cross the branch cuts.f ( z ) z2 1)F(z) is analytic in any one quadrant. Re f ( z ) 0 for all z(Recall :Re w 0x) Sommerfeld branch cuts27
Analytic Continuation Principle (cont.)Example (cont.)yDo we need another branch cutdue to the Ln function?(cosh 1 ( z ) Ln z z 2 11 1)xExamine argument of Ln function:1 1 w z z 2 1 z w 2 w ( derivation omitted)The branch cut for the Ln(w) function corresponds to w being a negative real number.Note : w ( , 0 ) z ( 1, )28
Analytic Continuation Principle (cont.)Example (cont.)yFinal Picture(cosh 1 ( z ) Ln z z 2 11 1)xF(z) is an analytic continuation of cosh-1(x) off of the real axis.F(z) is analytic in any one quadrant.29
EM ExampleAssume a radiating phased line source on the z axis.zNote: j is used here instead of i.I ( z ) I 0 e jkz zkz is realyxThe magnetic vector potential is µ I 0 (2) jk z zAz H(kρ)eρ 04j ( k02 k z2 )1/2 kρ k 2 k 2 , k kzz00 22 j k z k0 , k z k030
EM Example (cont.)We can also writeAz I0e jk z z ′z( x, y , z ) e µ0 4π R jk0 RR dz ′ I ( z ) I 0 e jkz z( 0,0, z′)yNote: The integral converges for real kz.xHenceAz jk0 R µ0 I 0 (2)e jk z z ′ jk z z′IedzHke ()µρ 0 ρ 0 0Rj44π Exists only for real kzExists for complex kzThe second form is the analytic continuation of the first form off of the real axis.31
EM Example (cont.)z µ0 I 0 (2) jk z z()ρAz Hkeρ 04j kρkz is complex( k02 k z2 )1/2 j (k k2zI ( z ) I 0 e jkz zy)2 1/20xIn order for this to be the analytic continuation off of the real axis of theintegral form, we must chose the branch of the square root function correctlyso that it changes smoothly and it is correct when kz real k0.kρ j ( k z2 k02 )1/2 negative imaginary number for k z real k032
EM Example (cont.)kρ j (k k2z)2 1/20Im ( k z )Im k ρ 0kρ j k z2 k02Im k ρ 0k0Re ( k z ) k0Im k ρ 0Im k ρ 0The Sommerfeld branch cuts are a convenient choice.(but not necessary)33
EM Example (cont.)kρ j (k k2z)2 1/20kρ j k z2 k02Im ( k z )(Top sheet)Riemann surfacek0 k0Re ( k z )Note:The function is continuouson the Riemann surface.We can now let kz wander anywhere we wish on the Riemann surface, and we know how tocalculate the square root. (We analytically continue to the entire Riemann surface.)34
EM Example (cont.)Example j (k kkρ 2zWhat is kρ at the finalindicated point?)2 1/20kρ j k z2 k02Im ( k z )Riemann surface k0Final point(Top sheet)Im k ρ 0k0Re ( k z )(Bottom sheet)Im k ρ 0At the indicated final point, the imaginary part of kρ is chosen to be positive.35
EM Example (cont.)Examplekρ j (k k2zWhat type of wave is atthe indicated points?Im ( k z ))2 1/20"Surface wave" : k ρ j ( )kρ ( ) , θ2 0)(θ1 π "Phased array" :Riemann surfacek0Re ( k z ) k0Note : k ρ j ( k z k0 )1/2( k ( k ) )1/2z0Leaky wave : k ρ ( ) j ( )(θ1 π 1 , θ 2 2 ) ; 1 2Note:The leaky wave field is the analytic continuation ofthe phased array field when kz becomes complex.36
Schwarz Reflection PrincipleAssume that f (z) is the analytic continuation of a real function f (x)off the real axis (or a segment of the real axis).Then within the analytic region, we havef (z*) f ( z ) *(proof omitted)yf ( z)Line segment: f ( x )Examples:sin ( z ) , e z , J n ( z )ln ( z ) , Re ( z ) 0xf (z) is assumed analytic in this region.(assuming branch cut on negative real axis)37
Analytic Continuation of Functions. 2 We define analytic continuation as the process of continuing a function off of the real axis and into the complex plane such that the resulting function is analytic. More generally, analytic continuation extends the representation of a function
COMPLEX ANALYTIC GEOMETRY AND ANALYTIC-GEOMETRIC CATEGORIES YA’ACOV PETERZIL AND SERGEI STARCHENKO Abstract. The notion of a analytic-geometric category was introduced by v.d. Dries and Miller in [4]. It is a category of subsets of real analytic manifolds which extends the c
Chapter 14. Eigenfunctions in the complex domain 339 14.1. Grauert tubes and complex geodesic ow 340 14.2. Analytic continuation of the exponential map 341 14.3. Maximal Grauert tubes 341 14.4. Model examples 342 14.5. Analytic continuation of eigenfunctions 343 14.6. Maximal holomorphic extension 344 14.7. Husimi functions 345 14.8.
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