DEFINITE INTEGRALS MURRAY MANNOS. Submitted in Partial Fulfillment of the Requirements for the Degree of. Bachelor of Science in Mathematics

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1 DEFINITE INTEGRALS by MURRAY MANNOS Submittd in Partial Fulfillmnt of th Rquirmnts for th Dgr of Bachlor of Scinc in Mathtics Massachustts Institut of Tchnology Jun 1937 Profssor in Charg: Profssor R. D. Douglass

2 INTRODUCTION Th first two chaptrs consist of th valuation and xtnsion of a numbr of dfinit intgrals; whras, in th third chaptr, an attmpt is d to show a possibl mthod for obtaining th sum of a finit sris by mans of dfinit intgrals somwhat of th form usd in Chaptrs I and II. It is prhaps advisabl to outlin th usful notions and formula which ar usd rpatdly in th valuation of th dfinit intgrals. This will nabl th radr to follow th stps mor radily without an undu rptition of th sam thorms, tc., btwn quations. Th notions and formula, th proofs of which y b found in practically any txt on Functions of a Complx Variabl, will mrly b statd as it is not dmd ncssary to rpat such proofs sinc this would mrly tak up spac. Mthods of finding rsidus: (1) Simpl pol as of function 1 f(z) = (z) z - a Thn Lim (z - a) $ (z) = z -a f (a) is th rsidu at a if th Limit is a dfinit numbr.

3 () It follows from L' t Hospital t s rul that, if W(z) and S(z) ar holomorphic at a and if z - a is a nonrpatd factor of V/(z), th rsidu of at a is _4 ial_ - (a) ( f h (a) (3) If w hav pols of highr multiplicity, thn w us Taylorts xpansion f (z) is ral or complx. (z -a) whr a (z - a) (a) + (z - a) $' (a) + (z - a) _a) 1 If n = 1 thn $ (a) is th rsidu at If n = If n = 3 If n = n " 1'(a) _! gn-1l (n-1). n na ] Two usful thorms for showing that I around R (whr R--> oo and r--o) = 0 ar th following: m SR f () dz = R--oo i (9 91) K around larg circl and lim z f (z) = K Z CO or r If lim (z - a) f (z) =k z-- a whr k is a constant, thn lim f (z) dz = (@ - 1 ) k r-- o

4 Th following usful formula ar givn: Cos z iz + -iz =, cosh z z + - z + sin z iz i -iz sinh z z - -Z sin i z = i sinh z = 1 cos i z = cosh z = - 1 sin 9 S 7T whr 0 z g 7r

5 TABLE OF CONTENTS Chaptr I Evaluation of Dfinit Intgrals.... Chaptr II - Extnsion of th Evaluation of On Dfirnit Intgral to That of Many Othrs Chaptr III - Sumtion of a Finit Sris by Mans of Dfinit Intgrals

6 Chaptr I EVALUATION OF DEFINITE INTEGRALS cos mx 0 x 4 + a 4 z. a, a[cos (7r+-k7TI + i sin jvr + k7q] z =- sin z cos (T--r - + 1~~ whr k = 0, 1,, 3 Considr zi 1 imz z 4 + a4 77 = a 4 dz z z 3 z 4 = a 4 i 577 = = a 4 I % Intgral around R = 0 for r im(r cos sin 9) RigdQ -ir - mr SR sin 9 - Q R -T -mr sin 9 0 R4 - a 4 R4 _ a 4 o 9R- mr - R a- < 4 - S r d < = 0 as R--> R -a 4 0 mr (R - a )

7 Rsidu at 0( t Rsidu at a 4 S7r Rsidu at a 4 im a i(cos T+i sin st) 4 4 4a 3 37T 37T) i(cos 3- + i sin 11) 4 4 4a 3 4 4a3 (- -+i - Adding rsidus, w hav 4 4 a s - -F ",- 4 a0 (As + i1) i -/ i-j g S-1-1) -1 (-+i J- --- F 4 a 3 4 a 3 ( i i + 1 i) ( -7 -() 4, -i J (-J) 4 i- - a 3 a sin (f- 4 +r)

8 0 i m x - 00 x 4 + a4 ý sin( (a +7C) dx = a 3 Oo cos mx -0 x 4 + a 4 co cos mx o x4 + a 4 sin (E + -- ) dx = a 3 qt sin (Sý + - ) dx = a s 0o sin mx dx 0 -So x+ 4 = -co x4 + a4 S00 x sin mx dx 0 x 4 + a 4 Considr Simz z dz z 4 + a 4 Intgral around R = 0 for m and a positiv R i m R (cos 9 + i sin g) iri 9 dg 0 R a 4 r -mrsiln 9 R T d s R -a R -a 0 T7 S------S R 4 - a 4 0 mr9 -mr sin 9 7T 9mR (R 4 - a 4 ) dq----r 7T 7 mr (R4 _ a4) 0 0 = -7TR- [1 - - R ] as R--7 4 _ a 4 R - a d9

9 Rsidu at OC S im Q Rsidu at a i v i (- + i ) E 4a i 4 a 1 Rsidu at a ( 1) + 4 a -4 a 1 Adding rsidus a Sx i mx x + adx = --- sin a si 7 i '3 a sin JE oo- x cos + amx 4 dx x + a o o oo x sin mx S x + a dx 17f a sin- JE o0 X cos mx - x x- 4 o- + a 4 dx = 0 o x- sin mx dx = 0 x 4 + a 4 a sin -- E

10 o00 x 3 sin mx S dx 0 x 4 + a4 4 z im dz z 4 + a dz z 4 + a 4 až 0, m 0 -R 7r R 3 3iG imr (cos 9 + i sin 9) S R i + a 4 R i d R 4 -mr sin 9 4 r7 < 7 R-- d9 Z -- S 0 R 4 - a 4 R 4 - a 4 0 A mrg dg Rsidu at o< Rsidu at a Rsidu at a Adding rsidus 4 R 4 7r R ---- R [ - - mr] - (R4.a 4) m R 71 4 r Sia im( 4 o( (+ i) r I (- -L + 1 A) F 0 as R - -, o i , 4 cos -- Sx3 i MX _- x4-4 dx = x + a Ti J cos JE

11 10 co Xx 3 s cos mx +i on x 3 s sin mx J o =i dx+i = 1dx 7 4 COS -cx S x 4 + a 4 x@ x4 + +aa4 o x 3 cos mx dx = -x + a0 dx x + a 0 o0 x3 x sin amxdx 4 =+ o00 X dx 0 x 4 + 5x + 4 cos -- dit Considr z dz z 4 + 5a za z = - 1 or - 4 z = + 1 or + 1 rsidu at pol G:4z IoCI <, K rsidu at = = 6 4 c\ + 10 S " i = ---- =- i 4 (-4) Sinc th xponnt of th numrator is lss than that of th biggst xponnt in dnominator (i.., (i.., z- th ; = 0as 0 R-> R-'>co) i \

12 - x dx 03o = i x 4 + 5x+ 4 sinc w hav an vn function i i So - x dx 0 x 4 + 5x x dx 0 x 4 + 5x + 4 i 6 7Fi 7/- -i 67 6 If M > 0 valuat soo cos mx dx x + x 4 considr imz dz z 1 + z + z = 6 z - 1 z - 1 whrvr 1 + z + z 4 = 0 61 = = 0 z = cos (0 + k-t) + i sin (0+-k6f) 6 6 (k = 0, 1,,, 4, 5) imz *"* + -z has pols at i th x axis th rsidu at ) i ( u, 3 abov i TT + th rsidu at 3 Lim z >-3 z ->9 im z z + 4z a3 Lim imz + i 7E 3 z + 4z

13 1 im ( + i t) i 3 (1 + ) Li -M ( )[1+( i))] -m 4 / i4x - i4 on th circl im (- + i -- ) 7 _E (1 + 3) -i -m [ ] (1 + 1 )[1+g(1 1))4] -m 7 1 -m i (1 + JE i) zj1 i T (1 - i JE) -i L ( (1 + 1 J4) (1 - i JZ) S_ (i + 1 4t (i z = R i 9 (- 1 + JE 1)(- JZ) dz = R i ig dg -m S --- cos ( JE, i 1 S imx coo S-m R sin im R cos 9 dx + +dq Ri R g + R 4 1 i i JE cos (7L - ) 3

14 13 th scond 0--O whn R -- co -m Cos x dx + i s sin mx dx = 7T cos (I -- ) -co 1 + x + x 4-00oo +x+x4 3 Scos mx dx 0 - +x-+- x 4 "0 i + x + jj -m cos- ( - ) 3 and S sin mx dx 1 + x + xi = 0 P S 0 a-i x-- dx i - x a-1 3 AL- dz z - 1 z = r i dz = r i i 9 dq 0 z-a 4 1 Lim r --> 0 0 r a - 1 i(a-1 ) 9 r i i 9 dq IT C r i 9-1 ra r a - ir r+ 1 - r + i 0 S d9 -- >' 0 Lim 7 Ra-i i(a-!)9 Lim S i R R->oo R i i9 d@ 9 R 0 R -1 7r Ra Ra E S 3' d- - 0 R+I R+I1 7r dg -- 0

15 14 i- a-1 S 0 x- 1 dx+ a-1 S x - 1 dx _ ari i- a-1 & dx+ 5' x xa-1 -dx = + f., x1 L,r 7(a-1) 1 + o (a-1) ( aiti - 1) P S 0 X a1-x -1 dx = 0 1-x 7T i ( r a i + ps xa-1 1-:x- a7ri I l1 (atri + -atri) a 771i - -a7r i i s nos a7 = sin a7 71ctn a7 00 sinh ax d 0 -sinh x 0 sinli 7w Considr az dz sinh 7T Z 7Tz - 7Tz sinc sinh z = 7-r - V whn 7w Z -7p Z w hav pols whn = z 1 i.., whn z = 0 and z = ki z = i nd only concrn us.using abov contour.

16 15 z = R+iy dz = i dy 1 iay 0 FR iy - -R -iy / a R 1 R - -R/ =- - a S dy tnds to 0 as R--> 0o z = -R+iy dz = i dy --- tnds to 0 as R --> S 0 ax a o0o sinh 7 x dx a(x+i) sinh Tr (x +i) 0 a(x+i) + snh dx co sliity + dx 0 ax dx + S -oo sinh -- dx -rx Lim z -> 0 z az 7 z - - zr Z Lim z-> az az + az 7 z + T --rz 1 (V ± _(_z_- ) az z - Z rsidu about littl smicircl -1-7T = i 7T Combining = -i i a i ) a oo ax ia) 0 ax 0 sih 7 x -7X sinh 7T (+ ia)s dx + (1 + ia dx + i-iia= 0 x

17 16 (1+ ia) s o ax (sinh -a 7rx) dx = - - [1 - i a ] 00 sinh ax d [1 - ia 0 sinh-rx [i + ia ] _ia ia - -i a + sin a _ 1 a cos - tan a S a - dbi cot z = iq COS 9 - a -bi Scos a - bi Ti ± 9-a-bl Tr d9 = -j. + d S dg = lu i - ia + b + 1 dq = i-ia+b i dq ia-b z + dz z b-- = - i z- i a - b z = - 7ri i = 7Ti If b / 0 If b ; 0 dz = i dg dz iz = dg ia-b is outsid unit circl ia-b ia-b is insid unit circl

18 17 S Lim z bz- - ýi z--o (z - ia - b) z T 9 - a- bi S- cot dg = -i whn b ; 0 rw = - 7ri whn b < a < Soo 0 xa dx 1 + x + x Considr a- 1 a dz dz Z (1 + z + Z 1 + z + z' z (1 + z + z9 ) z + z + 1 = Z z J3 Lim z = insid figur [ (_ 1 J+ i) ] z a z [z (-1 + )][z - ) 1) 3 Z v- I' Y4t. Intgratd around r = Ti lirm z z a = 0 z - Z (1 + z + z ) 0 xa-0i dx + S x+x a-i dx = 7iRsidu 7Ti.Rsidu 1 + x + X

19 18 co xa- 1 ri(a-1) xa-1 S -dx + S dx x + x' x + x V ST '3 Cos _Uq-a + i sin _-ffa cos + i sin - 3 (cos 83 + i sin g ( (cos 8L + j sin 8ELE)(cos?7 - j sin?th) (cos + i sin a-)(cos -7-1 sin ;_) -- JEa [cos (7Fa - )+ i sin (7 a 7)] o 1x a X+ Xx 0a-I cost a dx + S-- x----- dx = 0 1+x+x 77 - j- O 3,[) 3 i S 0 xa sin- a dx 1 - x +a i sin (7 ra - 7) 3 3 S a-1 x= dx x + x sin 7a sin a sin -r a 1 log (x + 1) --- x x log Considr l ( + z + 1 -~ dz S around R = 0, for lim z f (z) = 0 Z --, co

20 19 z_ ogl _ + -Z = z+i + log (z + i) z z 1 + -_z +- - ~z+i z = Rsidu at i = lim z-i_og_ + i) z.-> 1 ((z - i) ( z + 1) 1_o_ = i J sgx o di = x rr log i +1 r?) A-o -.- / / / --- v - 1 -

21 0 Chaptr II EXTENSION OF THE EVALUATION OF ONE DEFINITE INTEGRAL TO THAT OF MANY OTHERS Th following xampl will show from th valuation of on dfinit intgral how w can obtain a grat numbr of othr intgrals: oo a cos x + x sin dx -o00 x + ad Considr S -- z - iz - dz ai a ' 0 S0 77r ir(cos 9 + i sin 9) Rii9 do R i 9 - ai < r -RsinG ~ 7r R s ---- R R sin 9 0 R + a R+a 0 R+ a R+a 0 s - R w- S+ _-_R [1 - -R] 0 (R + a) R as R--> oo Rsidu at Limr z -->ia z -ai iz z - ai = -a oo ix Soo x -adx = 7ri -a -c0 x :-ai x + i sin x dx = Scos S ax S~ca i- at 7Ti - a o (cos x + i sin x) x + ail -a dx = Y-i -co (x - ai)(x + at)

22 1 o0 x cos x-a sin x + i 00 a cos x+x sin dx = i - a i-d 7.i -a 0o a cos x + x sin x - a () ' x---- a---dx = T () I d - oo x + a (3) S00 = 0 bcaus odd function x cos x dx - a 00 sin-x dx 0 i dx -a o x +a = x + a 0 So.co sin x x + a co x+ = 0 a' a > 0 +oo + -a cos x + x sinx dx = a x" + a Considr ix 0 0-0o x+ia iz dz z + ai dx = 0 0c0 cosx + i sin x (x - a -co (x + ia)(x- ia) = 0 oo x cos x + a sin x - a cos x + x sin x S dx+ i dx = a -c0 x + a (4) (5) co -_a-cos x + x sin x dx = d - o x + a 00 x cos x + a sin x dx = -00 x + a

23 Adding (1) and (4) (6) yo 0 x sin dx= 7 - a -0 X + a (7) 00 x-sin- dx = rs -- a 0 x + a Adding (3) and (6) (8) oo x + 1 sin dx = - a - 0o x + a Adding (6) and (8) oo (x sin x d-a So -dx = ar -o x + a Gnralizing, (9) oo nx 1 s dx = ntt -a whr n is an intgr -co X + a

24 3 Chaptr III SUMMATION OF A FINITE SERIES BY MEANS OF DEFINITE INTEGRALS Th schm consists of valuating --- x - dx 0 x n + 1 by a standard mthod; thn, knowing th answr, w compar this with th finit sumtions in th ral and iginary parts of an quation gottn by taking a numbr (n) of pols on th unit circl, whr w y lt ths pols approach arbitrarily clos by incrasing n sufficintly. 00 x p - 1a 0 p 1 C -- A-IF d 0 1 +x S P-1dz 1+ z Lim z f (z) = 0 RL 1? Lim z f (z) = 0 z-ž>o z--> 0 xp-1 0 (x _ p-1 oo dx + So -K- dx = iri ( p - 1)Ti S1 +x 1 + x Lim 1 + z z p - 1 p-1 sinc z (z-+ ) = ( ) OO p-1 0 pvri p-1 (p-1) ri IO -- dx dx = 7i x x

25 4 (1- p i dx = - 7Ti pi 0 + x 0 xp-i 0 l+x + v~i (- -Prri + p ) thn S x --d 1 thn X dx= T7r sin py 7rsin pit If m and n ar positiv intgrs, and m 4 n S00 xm +dx 0 x + 1 1AS -- S 0 m 1- o00 n dy 0 1+y (m+1 - ) y dy x n-1dx dx =y = dy 1- dx = Y... 1 X = y~n dy dy Sinc m ) (m+ i dy = 1+y - T S J sin I 771 xm sin m + xmn x + I dx m = yn W now attack th problm in a diffrnt nnr which will prmit us to gt th sum of a finit sris by mans of dfinit intgrals. W hav th advantag now of knowing th valu of th dfinit intgral dx Sm dx+

26 5 m and n positiv intgrs with m 4 n xm oo S dx 0 x + 1 zm S -n dz z n + I -A /~I~; z = - z = 1 (Cos I+ ktt + i sin L (k = 0, 1,... -1).'. w hav pols at angls t ' on unit circl Th pols from... - on unit circl ar n pols in uppr half of th plan and as long as thr ar no pols on th x-axis. Lim S R -- o0 AB f (z) dz = ( - 91) a b If f (z) ( V(z) n is finit whr Y (z) is a polynomial of dgr n and (z) is a polynomial of dgr lss than n; whr a and b ar th cofficints of z n - 1 and zn in (z) and /-(z) rspctivly. Sinc th dgr of $ (z) is Z-n -, a = 0 and.'. th intgral of f (z) round smicircl of radius R is 0.

27 R m Sdx = 7T i 'Y Rsidus -R x + 1 Taking pol at ) m P P1-1 ( ) p r Km+1j p --- L.Z n p -rr p =, 3, Substitut p = p - 1 p=1n p=l KLm+111p-1i7T n p=1 +1 (p-1)7 n p=1 (cos m + (p - 1)M7 + i sin m+t 1 (p - 1)7F) (A) S0 x x +1I d = 771 p=1 p=1 m + 1 (p - 1) 7 S p=7 p=ia sin m + 1 (p - 1) 7F sinc w hav an vn function oo m S dx n p=1 m + (p - 1) 7

28 7 0 xm x + n p=1 sin m+ 1 (p - 1) r sin m from comparison with th answr obtaind in th first mthod. Sinc cos m +1 (p - 1)7 = p=l for th iginary must vanish in quation (A) as w know from th answr obtaind in th first way. p=1 Chcking th rsults of this cos (ap + b) n i(ap + b) p=1 spcific finit sris: (only ral part of it) ib p=1 iap Using gom. sris n p=1 iap ia(n+1) ia ii -1 a S = i.*-1 _ a _- _- a ia - (m+1) - ia -1 i a sin a

29 8 '7 p1j ib ia sin (b + a) cos (ap + b) = sin -g sin a n p=1l sin m-+l (p - sin (ap + b) p=1 p=l 1 m + 1 sin -- 7n i(ap+b) sin s for (only ig. coff.) = b n p=1 ib iap = b i sin a i cos (b + a) - cos O sin a sin (r_+ 17T.- 1 sin 7-T W not th pculiarity of th finit sris n cos m ( - )7 = that is, no ttr how ny trms w hav in th sris givn by n th sum is always zro (i.., if n is a finit

30 th scond and fourth quadrants. 9 intgr no ttr how larg n is (with m < n) thn whrvr th sris is brokn off (at n = 5, 1000, 1010, or any such intgral numbr) th sum is zro. Chcking th rsult in spcial cass For n = For n = 3 3 p=1 Suppos for s~implicity M = 1 cos (p- 1)7F = cos + cos = 0 Z cos (p- 1)I = cos- + cos + cos5 = 0 6 p=1 For n = 4 4 cos (p- 1) 7/ = cos + cos - + cos p=1 + =I1 0 8 Sinc from th diagram and knowldg of sign of cosin bing positiv in th first and third quadrants and ngativ in A

31 30 37r 17C - cos cos -- T = cos --- r + cos = tc., for n > 4 (continuing th sam procss).