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2 2 http://www.ozawa.phys.waseda.ac.jp/inde2.html But nothing s unconditional, The Bravery > (, ( > (, f, g > f (,, > sup f( ( M <. > f(g( (i > g [, lim g( (ii g > (, ( g ( < ( > f(g( (i g < < f(g( g( f(

f(g( g( g( ( f( f( + f( 2Mg( ( > (ii ( d f(g( f(ydy g( f(g( [ f(ydy g( 2M ( ] f(ydy g( f(ydy g( + g( + sup << g ( ( f(ydy g ( ( f(ydy g ( f(ydy g ( (ii > f g g lim [, f( χ [/2, (, g( χ [/2, ( 2

f( { cos, /2, < /2 M, g /2 (i (ii n n k k2 (k k k k2 2 n k2 (k k k 2 n k2 k 2 n 2 log n (n (θ > θ ( θ < θ < 2 θ 2 (, f(, g( f θ g (i (ii < θ < 2 θ 2 θ ( < θ < θ < 2 ( θ θ ( 2 2 2 I. 2. F (t t e, t > 3

F (t t > [ e t e t (cos + t + t2 F ( F (t t F (sds t e + ] t t e + Tan t + t 2 + s 2 ds t Tan t 2 2. e t dt, > [, n] n n ( e t dt ( n e t dt 2 [ e t (cos + t + t2 + t 2 dt ] n dt e nt (cos n + t sin ndt + t2 e nt (cos n + t sin ndt + t2 n [, n] /2 n e nt e nt 2 cos n + t sin n dt + t dt 2 + t 2 e nt dt n 4

3. ( \ {} z eiz z D {z ; < z <, Im z > } D e iz z dz z Imz> e iz z dz + e i e iz z z dz + e iz z dz Imz> D e iz z dz i ep(ie iθ dθ i z Imz> ep(i cos θ sin θdθ z Imz> z dz e iz 2 ep( sin θdθ 2 /2 /2 ep( sin θdθ ep ( 2 θ dθ ( e sin θ 2 θ, θ [, /2] z Imz> e iz z z Imz> i + i z dz + z Imz> e iz dz z (ep(ie iθ dz z Imz> e iz z dz i ep(ie iθ dθ ep(ie iθ ie iθ ep(ite iθ dt e iθ ep( t sin θdt e i + e i e i + e i e i e i 2i 5

2 ( + 2 2 2 4. ( z eiz z D {z ; z <, Imz > } D D e iz dz z z Imz> z Imz> z Imz> z Imz> e iz dz + z e i e iz z dz z z dz + Imz> e iz z dz i dθ e iz dz i + z e i + e i + e i e i e i e i 2 2i z Imz> z dz 2 e iz 5. n D n ( e ij, j n n n D n ( + (e ij + e ij + 2 cos j j j 6

D n ( e in 2n k e i(n+/2 e i(/2 /2 sin((2n + sin((n + /2 sin(/2 e ik in ei(2n+ e e i /2 e i(2n+ e i(n+/2 e i(n+/2 e i e i(/2 e i(/2 ( + 2 n cos 2j n [ sin 2j 2 + 2 2j j j ] /2 (2n+/2 2 2 (2n+/2 /2 /2 /2 sin((2n + /2 f( sin((2n + sin((2n + sin((2n + f(, (, /2] f ( + cos 2 sin 2 sin2 + 2 cos 2 sin 2, 3 3! + O(5, cos 2 + 2! O(4 ( f(, f ( 2 f f [, 2] /2 f( sin((2n + : (2n+/2 [ ] /2 cos((2n + f( + 2n + /2 f ( cos((2n + 2n + /2 f ( /2 2 f cos((2n + ( 2n + ( sup f /2 ( [,/2] 2n + cos((2n + 2n + 7

II. ( 2 2. 2 (2 [ ] (2 + sin 2 sin y y dy 2 2 cos n ( 2 [ ] n (sin n 2 + (sin n2 n + 2n 2 cos ( 2 2 2 n ( 2 2 n + 2n 2 3 2n 2. ( \ {} z e2iz D {z ; < z <, Im z > } D e 2iz dz z Imz> z Imz> e 2iz dz + e 2iz dz i e 2i 2 e 2iz + z Imz> ep(2ie iθ dθ e iθ D e 2i 2 8

z Imz> e 2iz dz e 2iz dz z Imz> z Imz> Imz ep(2ie iθ dθ ( + ep( 2 sin θdθ 2 2iz dz z Imz> 2i z z dz 2 dθ 2 Imz> e 2iz 2iz 2iz z Imz> e 2i + 2 2iz 4 e 2iz 2iz e 2i 2 + e 2i 2 2 e 2i e 2i 2 e 2itz 2iz (e 2itz dt dz e 2i 2 e 2iz 2iz dz e 2istz dsdt 4 z 2 4 dθ 4 (e i e i 2 4 2 ( 2 2 + 2 ( 2 2 2 ( 2 9

3. ( z +2iz e2iz D {z ; z <, Imz > } D D + 2iz e 2iz dz z Imz> z Imz> z Imz> z Imz> + 2iz e 2iz dz + + 2i e 2i 2 e 2iz dz + 2i z z dz + + 2i e 2i + 2i e 2i + 2 2 Imz> e 2iz 2i e 2i + 2i e 2i dz 2 dθ + + 2 2 e 2iz dz 2 + 2 e 2i e 2i 2 ( 2 2 4 z Imz> e 2iz dz 2 4. sin 2 z 2 sin 2 z z /, Z 2 sin 2 n n n (z n 2, z \ Z (/ n 2 sin 2 ( n 2 n sin 2 ( n ( n 2

(, < < /2 { sin 2 ( n, n ((n + sup 2 [, ] ( n 2, n (n 2 { sin 2 ( n lim ( n 2 { sin 2 ( n lim ( n 2, n, n, n, n n sin 2 ( n ( n 2 [, ] [, ] n sin 2 ( n ( n 2 n (n+ n sin 2 y y 2 dy sin 2 2 III. 2. < p <, >, t > u(t t u (t i i p t p ep( + it p ep( + it p e d [ t p e e it ] p t u(t i t u (t ( it eit t p e e it + t d (p e e it p e e it

u (t p t + i u(t ( d p (ep(p log(t + iu(t ep(p log(t + i dt t + i u(t + u (t ep(p log(t + iu(t ep(p log(iu( (i p u( (i p p e i p Γ(p u(t Γ(pi p (t + i p p /2 t, cos ( Γ 2 2 2 2. ( 2 e t2 dt, > [, n] n 2 n ( e t2 dt 2 ( n 2 2 [ e t2 dt e t2 + t 4 (cos + t4 + t dt 2 4 ] n dt e nt2 + t 4 (cos n + t4 sin ndt 2 2 e nt2 + t (cos n + 4 t4 sin ndt n 2

[, n] /2 n 2 2 e nt2 + t cos n + 4 t4 sin n dt 2 e nt2 dt n 3. ( t /2 e t dt, > n n ( t /2 e t dt n ( n t e t dt [ e t (cos + t + t2 + t 2 dt t ] n t dt 2 e nt + t (cos n + t sin n dt 2 t e nt + t 2 (cos n + t sin n t dt n ( 4. z ep(i D {z ; < z <, < Argz < /4} D [, ], {z : z } {e iθ ; θ /4}, {z; Argz /4} { +i 2 ; } ep(i dz ep(i 2 + i /4 ep(i 2 e 2iθ e iθ dθ 3 ep ( i / 4 D ( 2 + i + i 2 2

/4 i ep(i 2 cos 2θ 2 sin 2θe iθ dθ /4 ep( 2 sin 2θdθ /4 ( ep 2 2 (2θ dθ [ ] /4 ( 4 ep 42 θ 4 ( ep( 2 ( ep i 2i + i 2 2 + i ep( 2 2 2 ep(i 2 + i 2 ep( 2 4 ep(i 2 + i 2 2 sin( 2 cos( 2 2 2 cos 2 émi arles, Due eercises d intégration, Notes de cours http://www.math.univ-montp2.fr/ carles/ 934 I,II 4

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