Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas)

Transcription

1 Lecture 7: Analytic Function and Integral (See Chapter 4 in Boa) Thi i a good point to take a brief detour and expand on our previou dicuion of complex variable and complex function of complex variable. In particular, we want to conider the propertie of o-called analytic function, epecially thoe propertie that allow u to perform integral relevant to evaluating an invere Fourier tranform. More generally we want to motivate the proce of thinking about the function that appear in phyic in term of their propertie in the complex plane (where real life i the real axi). In the complex plane we can more uefully define a well behaved function ; we want both the function and it derivative to be well defined. Conider a complex valued function of a complex variable defined by F U x, y iv x, y, (7.) where UV, are real function of the real variable xy., The function F i aid to be analytic (or regular or holomorphic) in a region of the complex plane if, at every (complex) point in the region, it poee a unique (finite) derivative that i independent of the direction in the complex plane along which we take the derivative. (Thi implie that there i alway a, perhap mall, region around every point of analyticity within which the derivative exit.) Thi property of the function i ummaried by the Cauchy-Riemann condition, i.e., a function i analytic/regular at the point x iy if and only if (iff) U V V U, x y x y (7.) at that point. To ee thi reult tart with the fact that F i a function of and then apply the chain rule in the x direction and in the y direction to find F df df df F U V i, x d x d d x x x F df df df F U V i i i y d y d d y y y. (7.3) Phyic 8 Lecture 7 Winter 9

2 By equating the real and imaginary part of the expreion for df d we are led to Eq. (7.). For example, the function F x y i xy,, V xy, atifie U x y the C-R equation at all, while F x iy atifie the C-R equation nowhere. Next conider the function F x iy x y x y, (7.4) for which x U x y x,, U x y x x y x y x y U xy y x y y V xy V y x V,,. x y x y x y x y, (7.5) We ee that thi function i analytic everywhere except the point where it i aid to be ingular (both the function and the derivative diverge). Unlike point of analyticity, ingular point can be iolated point. In the (imply connected) region R of the complex plane where the function analytic/regular it ha many truly important propertie that we ummarie here. F i For an analytic/regular function (that atifie the C-R equation) the real function UV, are harmonic function, i.e., they atify Laplace equation U V. (7.6) The integral of an analytic function i independent of the path of integration in the complex plane a long a the function i analytic/regular at all point along all path conidered and between all pair of path, i.e. an analytic function i Phyic 8 Lecture 7 Winter 9

3 holomorphic in the region R. The integral of an analytic/regular function along a cloed path vanihe a long a the function i analytic/regular at every point along and inide the path. Let F be analytic/regular everywhere in a region R in the complex plane and let C be a imple (i.e., doe not cro itelf), mooth (with, at mot, a finite number of corner) cloed path entirely inide R. Then we have CR, F d (7.7) which i called Cauchy theorem. (Thi cloed line integral i called a contour integral in the complex plane.) For our purpoe the related but more relevant reult i Cauchy integral formula, F F d, i (7.8) C R where the contour encloe the point in the counter-clockwie direction and F i analytic/regular everywhere in the region R. Note that there i a ign change if the point i encircled in the clockwie direction. We can undertand the factor if we conider the pecial cae F and define C to be the unit circle about. For on thi path we have have F i e, d i ide. Thu we i ide d. i i e (7.9) CR F i analytic/regular at, the The fundamental point i that, although integrand F i not. It ha a imple pole at and the contour integral around the point erve to elect out the coefficient of thi pole, F, alo called the reidue of the integrand at the pole at. Phyic 8 Lecture 7 3 Winter 9

4 F i analytic/regular, all of it derivative exit and we In a region R where can write a Taylor erie expanion about any point R, F F n n n n! n! F F d, n i CR n, (7.) where i inide the contour C. If we ubtitute the firt expreion into the integral in the econd, we ee that only the term proportional to, i.e., the imple pole ingularity, contribute to the integral. Thi i jut like the previou reult. The circle of convergence of thi erie expanion about the point i given by the ditance from the point to the cloet point on the boundary of R, i.e., the cloet point to, where the function F i not analytic/regular, i.e., where it i ingular. ASIDE: Thu any function for which we know that the power erie expanion i convergent everywhere, e.g., e, in, co, inh, coh, mut be analytic/regular everywhere in the (finite) complex plane. Thee function are often labeled a entire function. Next we want to conider the erie expanion of the function about a point where it i not analytic/regular, a Laurent erie. Call the ingular point and aume for the moment that it i iolated, i.e., that there i a region R around (preumably in the form of an annulu) where the function i analytic/regular. We can till expre F a a erie expanion about thi ingular point, but now we mut include negative power (the an, n here are the ame a the b in Boa), n Phyic 8 Lecture 7 4 Winter 9

5 n F a a n F n i n CR n, d, (7.) where the integral defining the coefficient in the expanion i eentially identical to that for the previou Taylor erie. The contour C i again entirely in the analytic/regular region R, but now encircle the ingular point in the counter-clockwie direction. Note the pecial cae n, a i CR F d. (7.) Thi term i jut the reidue mentioned earlier and i the path integral of the function around the contour (divided by i ). Conider now ome terminology for pecial cae: if a, n n, the function if an, n, a, the function if an, n, a, the function if a, all n n the function think of the function e F i (in fact) analytic/regular at ; ). F ha a imple pole at ; F ha a double pole at ; F ha an eential ingularity at (e.g., In any of thee cae, if the point i the only ingular (non-analytic/nonregular) point encloed by C, the integral of F about the cloed contour C i given a above by ia, i.e., i time the reidue at the ingular point. If the ingularity i jut a imple pole we can alway find the ingularity via a lim F. For higher order pole more care mut be taken. Next conider what happen if more than one ingular point i encloed by the cloed contour C. Noting that we can alway move the cloed contour around by adding and ubtracting cloed contour with no ingularitie inide, which Phyic 8 Lecture 7 5 Winter 9

6 have ero integral from above, we are led to the Theorem of Reidue - df i Reidue, (7.3) CR pole inide C where we mut be careful about the ign. We add the reidue at ingular point encircled in a counter-clockwie ene and ubtract thoe encircled in a clockwie ene. For example, if F i ingular at,, 3 inide C with reidue a, a, a 3 and C encircle, in a counter-clockwie ene and 3 in a clockwie ene (omewhat hard to imagine, but till), we have df i a a a 3. (7.4) CR Now let conider ome example of uing the Reidue Theorem, Eq. (7.3), to perform integral. There are 3 baic tep:. If the integral i not yet a contour (cloed path) integral (in the complex plane), complete the contour by adding another path integral.. Find all ingular point inide the contour and evaluate the reidue at thee point. 3. Sum up the reidue with a + coefficient for CCW encloure and - for CW encloure. Multiply by i. Comment: The tricky part i tep., ince there i no univeral recipe for how to proceed. It i clear that the path integral along the added path mut either vanih (the uual cae) or be eay to evaluate. The latter ituation will typically arie due to the ymmetry propertie of the integrand (ee the firt example below). A typical added path with vanihing contribution i an arc at infinity in the complex plane where the integrand vanihe when the magnitude of the complex integration variable become large. Of particular interet are complex exponential factor in the integrand, which will tell u the ort of arc that i called for (ee below). It i often neceary to break the integrand into a um of term and cloe the contour for the different term in different way. Care mut be taken when branch cut are preent a decribed in Example 5 in Boa and exercie 4.7:33 in Phyic 8 Lecture 7 6 Winter 9

7 Appendix B. Step. i traightforward (well defined) mathematically, e.g., find the Laurent expanion, but may involve ome arithmetic. When only imple pole are preent, a i typically the cae in phyic problem, we can ue the relation a lim F. Step 3. i jut more arithmetic except that care mut be taken to correlate the ign of the coefficient with the direction of the path around the ingularity (ee the example) A a firt example conider I dx dx dx, x x i x i i x i x i (7.5) where we ue the ymmetry of the integrand to extend the integral along the entire real axi (with a factor of ½) and then factor the denominator to expoe the pole. Now come the eential trick. We chooe to cloe the contour by adding a line integral along a emi-circle of radiu R, connected to the real axi at both end, and then take the limit R where thi added path integral vanihe ( Re i along the arc), I Lim d Lim arc R R i R e arc i LimR ide. R id Re i (7.6) Here the in the limit of the integral ( ) remind u we could put the emicircle in either the upper half-plane (Im, ) or in the lower half-plane ( Im, ). The only difference i which pole we encircle and the ene of the encirclement. If we cloe in the upper half-plane, we have Phyic 8 Lecture 7 7 Winter 9

8 d i I I Iarc a i Cupper half-plane, CCW i i Lim i. i i i i (7.7) Here we noted that, ince there i only a imple pole at = i, we can ue the expreion above, a lim F. An alternative, and often impler approach, i to ue the partial fractionated expreion in the lat verion of Eq. (7.5), which tell u by inpection that a i i and a i i. If we cloe the contour the other way in the lower half-plane ( Im, ), we pick up the other pole and encircle it in the clockwie ene, d i I I Iarc a i Clower half-plane,cw i i Lim i. i i i i (7.8) Note that with either choice we obtain the ame reult a required. A long a we cloe the contour correctly, we mut obtain the ame reult. An example cloer to our goal in thi coure i to evaluate the invere Fourier tranform of the particular olution of the nd order differential equation, ax bx cx F t, where the right-hand-ide i an impule, F t C t. (Nothing happen until we hit the ytem with a hammer at t.) The Fourier tranform of the driving function i trivially found to be F t C t G C. Thu we can write (ee the end of Lecture 6) the particular olution a Phyic 8 Lecture 7 8 Winter 9

9 it Ge xp t d x d a i i it Ce d a i i it Ce d a i i b c b i, i i. a a 4a, (7.9) To perform thi integral uing the Reidue Theorem we mut cloe the contour uing the idea dicued above. In particular, we mut deal with the exponential factor it itre tim e e e. In order to cloe the contour with a emicircle at infinity we mut enure that the abolute magnitude of the integrand vanihe a R (formally thi rule i called Jordan Lemma). The econd factor in the complex exponential i a real exponential, which we mut enure i a damping exponential, and not a divergent one. So in thi cae we are not free to chooe to cloe in either half-plane, but rather thi retriction produce the following correlation: requiring the product t Im mean that A) if t, we mut require that Im and thu we mut cloe in the upper half-plane; B) if t, we mut require that Im and thu we mut cloe in the lower half-plane. Noting that i,, ib a c a b 4a or, i, we ee that the two pole in the complex plane are both in the upper half-plane (thi remain true even if the ytem i over-damped) and we have Phyic 8 Lecture 7 9 Winter 9

10 it Ce xp t d a i i it d Ce t a upper half-plane C C it de a it d Ce t a C lower half-plane t t ice e e a t i t i t Ce a bt a in t bt a Ce in t t. (7.) a Note that the turn-on of motion at t (i.e., the preence of the explicit theta or tep function) i automatically handled by thi method of evaluation! The particular olution automatically know that it exhibit no motion before the hammer hit at t. Thu the combined mathematical magic of the Fourier integral tranform and the Theorem of Reidue (analytic function magic) enure the correct time behavior (uually treated with initial condition). ASIDE: To practice our facility with the function of mathematical phyic, let u explicitly verify that the reult in Eq. (7.) i a olution of the initial equation ax bx cx C t. By explicit differentiation we find Phyic 8 Lecture 7 Winter 9

11 bt a t t Ce C e e xp t in t t t a ia t t t t t t t t t C e e C e e x p t t t ia ia C e e C e e t ia ia t C e e ia t t t t t t C e e C e e xp t t ia ia t t C e e C t ia ia t t C e e C t ia a In implifying thee expreion we have ued the knowledge that the derivative of the theta function i jut the delta function (ee Chapter 8. in Boa) and that the delta function can contribute only when multiplied by a function with a nonero value at the point defined by the argument of the delta function (t = in the preent cae), and we might a well evaluate the coefficient of the delta function at that value. Thu, when we ubtitute thee expreion into the LHS of the differential equation, we find t. t t t t t t C ax p bx p cxp a b c e a b c e C t ia C. The two quantitie in the curly bracket vanih due to the definition of the exponent,, a, b, c. Thu we have indeed found a particular olution of the initial equation. Phyic 8 Lecture 7 Winter 9

12 A a final example of thee technique let u return to the Fourier tranform in Eq. (6.3) in the previou Lecture and confirm that we can ue our new contour integral technique to perform the invere tranform. Thi will allow u to dicu another example of how the choice of (how to cloe) the contour define the final t dependence and allow u to dicu the cae when the integration contour pae through a ingular point. Thi lat iue i dicued in Boa Section 4.7, Example 4 and in exercie 4.7: in the HW. The central point i that the contribution of the pole on the contour i ½ what you would normally expect, but the ign till depend on how the contour i cloed. Recall we had the initial function in the figure and the invere Fourier tranform F t it in F t dge d e, t., t it (7.) To explicitly evaluate the integral uing our new technique we firt reorganie the integrand to put it explicitly in the form in Example 4 in Boa. We have i i e e F t d e i it (7.) i t i t e e d d. i Note that in thi form we are explicitly integrating through the imple pole at the origin, i.e., there i a ingularity on the contour, and we need to be a bit careful. We can obtain the general reult from exercie 4.7: in the HW. In the cae that it i appropriate to cloe the contour in the upper half-plane we have Phyic 8 Lecture 7 Winter 9

13 upper half-plane dx x f f d i reidue of x x Im f x f f dx lim R d lim d x f dx lim id f Re x R n R n lim id f C n re n! r C f i dx lim id f Re i f. x f x r i in f i reidue of i f Im i lim id f Re, R (7.3) where we are thinking of the contour in Figure 7.3. Note that the pole at the origin doe not contribute explicitly (it i outide of the contour), but it doe guarantee a nonero contribution from the vanihingly mall contour C. Thu for the imple cae of Example 4 in Section 4.7 of Boa, where the numerator i jut the complex exponential telling u to cloe the contour in the upper half-plane (where y give a damping exponential and the integral along C vanihe) and there are no pole except at the origin, we have (f() = ) ix e dx i. (7.4) x If the ituation require intead that we cloe the contour in the lower half-plane (but with the mall emi-circle C the ame), we now um over the reidue in the lower half-plane with the oppoite ign (CW direction), include the contribution of the reidue at the origin, which change the ign of the econd term, and the lat integral along the contour C i from to - (alo a change of ign). Thu we have for the lower half-plane Phyic 8 Lecture 7 3 Winter 9

14 Im Im f x f dx i reidue of if x i limr e i f id f R f i reidue of if i lim id f Re. R (7.5) (Note that, if we alo flipped the mall emi-circle C into the lower half-plane and did not encircle the pole at the origin, we would till obtain the change in ign due to the changed integral along C,.) So in the cae of the complex conjugate of the imple exponential, which i to be cloed in the lower half-plane, y <, we get the expected complex conjugate reult (compare to Eq. (7.4)) ix ix e e dx dx i. x x (7.6) So now we apply thee idea to the two imilar integral in Eq. (7.). For the firt exp i t we are led to cloe the contour in the upper integral with the factor half-plane when t > - ( Re i t the econd integral with the factor exp i t ) and in the lower half-plane for t < -. For we cloe in the upper half-plane for t > and in the lower half-plane for t <. Thu via contour integration technique we obtain that Phyic 8 Lecture 7 4 Winter 9

15 t t i t i t e e F t d d i t both in lower half-plane, i i i t t in upper, nd in lower, i i i tboth in upper half-plane, i i i. (7.7) So again the tructure of the contour integral enure the correct theta function in the t dependence. In particular, recall that the definition of the theta function mean, t, t t,, t, t, t, t t, t, t, t t t, t., t (7.8) In ummary, at leat in principle, we can now olve any nd order linear inhomogeneou differential equation (with contant coefficient) whoe driving function can be expreed in term of a Fourier integral tranform (or a Fourier erie, if the function i periodic). The only eential limitation i that we have to aume that the function i (abolutely) integrable. We will addre that limitation next uing the Laplace tranform. Phyic 8 Lecture 7 5 Winter 9