From 95752d21d5d98b2cea1c53b47453e8592af1f07d Mon Sep 17 00:00:00 2001 From: christian Date: Mon, 23 Dec 2019 19:59:35 +0100 Subject: [PATCH] remove enumerate from proof --- ws2019/ana/lectures/analysis.pdf | Bin 417631 -> 417662 bytes ws2019/ana/lectures/analysis18.tex | 33 ++++++++++++----------------- 2 files changed, 14 insertions(+), 19 deletions(-) diff --git a/ws2019/ana/lectures/analysis.pdf b/ws2019/ana/lectures/analysis.pdf index 460f0b09d4b46f5ac256f3c1e9749cd5752cc409..7822bc8627fb5fcce6d1bcf43ad016bfcb5330b5 100644 GIT binary patch delta 3987 zcmV;E4{Y$?z8U_$8GwWVv;q&i0y#F9ahU-qf4y2wa}!4pz3W%(Q7%fuOi#}jDGnhq zU`U(FT*WGV=b$j{bS5>^Zn91Tz zA&UpId&}AF`^FSO1WUG9zAl)Ap~5RIY$%qmif3~cM+$^Z`1rOsoBdXwp!%Cw zbSe~%y^z`tnUsZgLRr-^34~y#Wilw=e>55Y&)N&)bYk2gRa(G zZB}q&g#!<2BhRIj>orYB<);aL>NWp#Z2&J-eSP<6LC((!>;=tVzQ5QQY%JF#f0S|Y zDc9bal7$t)X&=`a;jpy@l7~!kM%P%x!YEfczm>6Kg;F-ze3~=LMt&gd&H-{W6n4cT z5Z6>H!`$49ax6SMm{P)pO~z9f040TGE|d~Rni>L?37{3u7mEZb)x{s%@~RudZV2~> zTR33eope>VNF6Olmqm;^h%&cois6NT3Zcw7)iW|u3sOioOi z>rB5M9|$kDJvIHh+rYWaHU;wgKoGe0LD{pYU>Y`ViiUPD^Zh}Y?=}1&^DTJe56gTX zP}o615XT!oocVf9L3?wVUwV|)zMf0UfsP0>IB$fh#L5o2AJ$vZYeP&_e`pkd{_JN) zY=OK*7w@)O9ctvU{nB3pc-!z54S94_@I`1(4d7c2(R8ez6l$E+=+t8qP%=_@ioG1q z8LA;wNjKkibt{{dg`^i+hGGHX!fF$cd#lf@^NY3|NCX7*Mlf~$DX|udKc9ch6bvoP z{<>Ma%7hx=17f6?+SI}Gf2%IHB3I=)3#(IS5Jhe9ot_NEU7s@=^atc)M$H>3XRf8^weYCvHL4Trgs zs7nj-I$KcN00ottx&w@bIz=aF943oN(&MLLa+TK#l0)LI(V$dc>zU>-6ZVCR>4M}^ z$V=mrG82Q+Q)(dD!?m$-LRyU8imq}tY~W^W2hM}r9{YjpK(jFRMoZ_U6vYxmPJ^Oa z;{zzcd_A2)Z=2l?f2z2pSR0u2tM5a@9QVq5T%el^7Lbpq0}GUkN;LhK(W?PDf{twC zG(pyr6YbHU3svg#&Df+XTA&}S+CEyk`1earm_d=D-t6PU8=<^UeSBRX7UgxY5;N`;@!Pb4|iK zcQb7j9Ag*ze=4*2hIiwL7jCLn<8a;{bT%+**6)|+#E7iWlWewn$N ztS@^LVd8H)$-=D}msp+z)yO1EnRrdNS*%QVlGVUUWDte+_d}W;RNGdVaZ&jxtFyYp z*Iu1{80>eGaFZ^uhcJhBJrn36K@+I2wL;eI z5ahc*Z%NU&UBAOTc8XxR%nFw6aCuy?s9g~5+44?rqb~Epe+SM?RfO3b&|5p{M2OoH zAs(Stf1+Y{NS9QRKu3j9sTKtjuFuA^aIR8OC=bvRV?s@qQfu>Mzfdz#JniGaB6h(f z4!(BPO{ut4h~x2KtC_);m5B`SfW!{qsuWDQ*qJR;P4n2HdTBl(Gu>`)O8SUnOT9Zz zoG?NhJ3<_S#>eSLh{FJhsZE?1F&b0fM9_$%f4pI>ga^y_a8SPF_c`JudM`4E9Yu9t zy@uby>XMfsW}9x?@{mvqTXI7kE|go2BUokHErF0C@kP->bGh-scjuXx=fg#b_b8#Q z=5BcGjefE#joWAh+VS8}Zn|Q~y)<%OIMRgycE$|;=!4?qmQtbZlUW7Q2GYWyqLJ49 zf3H2&`o?a{mrMsXE{K2cx)@CngFE4*p-#2B7MyfR&ydHdb+|Z6Xy9}wli%_85wU=P z#Kx%MU)b0(*6V#WO#nqg;E?dr4Zg@T$W&V&EoXwno))|kBzgC9^#4GJ3`7E%sJngj z;{-#=8$m$%KkO}#_zGoiWOH8IWRMq zoHhm@1T{G@Gq=As2C5)ZH#Im!I5##oH#s*pH#s*lHbyr@IXFc^GeR^oH#tT$J|H|d zH8?~#H#RpnIX5;pIX5ykMmIz`I7LD;LNqfsIYu--T?#KuWo~D5Xdp2-I5L-UnE@z& zCDwU-7R4C{@OgITec^=wA;b_uNFWJG2mz810)&K+69^=PaNh~2prRJ7;)NisVlCi> zAW*DT6sdwmN~zejTF%HRa^G^bf*e|*SP=dFp8WIu?6W&NyEF4VvulhoKYPabLDGOU z1R)_tjA>?gt|#dx-YbhV9LB4bcw!`fT+d~aPNV~APlm8bfH*P1Yuc_YDJQK-6{%no zK0Fi@6ERrZ4J6fMC>g}7A<|GxTzfVNH71S3beybr^e5pw*Mx+L>Ap}e3}@JkG!;{{ zT<>j8!o^g~(Xa)H5Hn_^hIf!iF?VHZ*pftvnO3V|D-tbcR=jpUf*}VJiV-t^Z;&== zLt@1&eoLDSU>Hy0#H`q==PDVtBW=a3y`;A#k_0iIy{RE5F_a```!Ky@B*PSvEau=i zZNkY8rHVOoOT&&NP0ZDF4Y?knbTPMwY6p!Nb|IZbn#5`ou3IQWB(|$I8O5+G$r9;! zK$~fM(U1!t>L>Df5AA*oL!I$*kzMu6tr<2@ z;Ghl&@qK)WByP}h98cnNENw298zNb#~> z)KK4Gw0K2_H5^CAidR|R{RA>zya_WktR*$#Jv3dr*T$2`MDb>?({rpw--x$lsgYp! zsd|p(=X>J4rwQz{%6uo@h8r59fd91ycWhh;HVp-v4Y>}sdewdlVJNf zu;Ucixex4G4|e|<>{$eVzPuOgeFA({4ffpu_BSJS;J{zN!KcBYyTRe{;7A8>v<=~N zkDb+5W7Ya$gk|bCCC$V;eicnlW`I)Xq_}{1C#!c}5OmH)QAKZEl{KO`=PZ+T# z97sIyFc@oR0XGK(-UPw7L4zHj;RVnLL$!^ob%5Mj{0PEb#g8Q1Q2Z#uy~K|u+(!Hu zv0;zuZLx&=h95`b#Wop>PMdB3*ww@~PXiH;fEHLm8;OOoQH{xWpym4@`T%IfmyE&u z+1O5?4OYg+od<1y8-R8JkT4r0_5w+(K{D3Trf{gK_kj+VKpMNp)*-eNN8V)s$XEd~ zd7*0*$i}SMZoDJs4S?NNY)?aOfLu;auOC4ka?R!+1iili1z2fYm;{R0OK}+}!TQ=# zzEIhnpf6I&_IIGX1`K!x46Fhb9A)KgP?Z7(b2h4X>HKkjEA3 zNAB7Ye*h!-2BVPZcJvuAwlx^X#T%akCZvLzSaJl^qBQJ8B%z(eSDVC@z7zRp??NWp zyLn-9BACK)OxXSLO4F?b31AdKcw~wG^?4z8f z>Bv9(Sa0xucs6)~9XvStS9MNw%j%S%A&qaVaL%{Q#@fVoI zzh@rIvb)Y3BPoHs@M;?jB9GaN7(=DTITa#Irrp8%`b1)xI0vo1+Pgp(wSPOh? 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Mit $e^{ix} = \cos x + i \sin x$ folgt direkt + \begin{align*} + \cos (x+y) + i \sin (x+y) &= e^{i(x+y)} = e^{ix} \cdot e^{iy} \\ + &= (\cos x + i \sin x)(\cos y + i \sin y) \\ + &= \underbrace{\cos x \cos y - \sin x \sin y}_{\text{Re}} + i \underbrace{(\sin x \cos y + \cos x \sin y)}_{\text{Im}} + \intertext{2. Setze $u := \frac{x+y}{2}, v := \frac{x - y}{2}$. + $x = u + v, y = u-v$.} + \sin x - \sin y &= \sin (u+v) - \sin (u - v) \\ + &= \sin u \cdot \cos v + \cos u \cdot \sin v + - (\sin u \underbrace{\cos(-v)}_{= \cos v} + + \cos u \cdot \underbrace{\sin(-v)}_{- \sin v}) \\ + &= 2 \cos u \sin v + = 2 \cos \frac{x+y}{2} \cdot \sin \frac{x - y}{2} + .\end{align*} \end{proof} \end{document}