From 271424a25ba50429c6a1f0dd35815465d431b2b0 Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Tue, 18 Feb 2025 12:28:22 -0600 Subject: [PATCH] update and fix typo --- pages/Math416/Math416_L10.md | 10 +- pages/Math416/Math416_L11.md | 121 ++++++++++++++++++ pages/Math416/_meta.js | 1 + .../Cauchy_theorem_triangle_carving.png | Bin 0 -> 83474 bytes 4 files changed, 127 insertions(+), 5 deletions(-) create mode 100644 pages/Math416/Math416_L11.md create mode 100644 public/Math416/Cauchy_theorem_triangle_carving.png diff --git a/pages/Math416/Math416_L10.md b/pages/Math416/Math416_L10.md index f2e3fec..b1cf4f2 100644 --- a/pages/Math416/Math416_L10.md +++ b/pages/Math416/Math416_L10.md @@ -40,7 +40,7 @@ EOP ### Complex Integral -#### Definition 4.1 +#### Definition 6.1 If $\phi(t)$ is a complex function defined on $[a,b]$, then the integral of $\phi(t)$ over $[a,b]$ is defined as @@ -48,7 +48,7 @@ $$ \int_a^b \phi(t) dt = \int_a^b \text{Re}\{\phi(t)\} dt + i\int_a^b \text{Im}\{\phi(t)\} dt $$ -#### Theorem 4.3 (Triangle Inequality) +#### Theorem 6.3 (Triangle Inequality) If $\phi(t)$ is a complex function defined on $[a,b]$, then @@ -74,7 +74,7 @@ Assume $\phi$ is continuous on $[a,b]$, the equality means $\lambda(t)\phi(t)$ i EOP -#### Definition 4.4 Arc Length +#### Definition 6.4 Arc Length Let $\gamma$ be a curve in the complex plane defined by $\gamma(t)=x(t)+iy(t)$, $t\in[a,b]$. The arc length of $\gamma$ is given by @@ -117,7 +117,7 @@ $$ \int_{\Gamma} f(\zeta) d\zeta=\int_0^{2\pi} f(\gamma(t))\gamma'(t) dt=\int_0^{2\pi} \frac{1}{Re^{-it}}iRe^{it} dt=2\pi i $$ -#### Theorem 4.11 (Uniform Convergence) +#### Theorem 6.11 (Uniform Convergence) If $f_n(z)$ converges uniformly to $f(z)$ on $\Gamma$, assume length of $\Gamma$ is finite, then @@ -144,7 +144,7 @@ $$ EOP -#### Theorem 4.6 (Integral of derivative) +#### Theorem 6.6 (Integral of derivative) Suppose $\Gamma$ is a closed curve, $\gamma:[a,b]\to\mathbb{C}$ and $\gamma(a)=\gamma(b)$. diff --git a/pages/Math416/Math416_L11.md b/pages/Math416/Math416_L11.md new file mode 100644 index 0000000..c736429 --- /dev/null +++ b/pages/Math416/Math416_L11.md @@ -0,0 +1,121 @@ +# Math416 Lecture 11 + +## Continue on integration over complex plane + +### Continue on last example + +Last lecture we have:Let $R$ be a rectangular start from the $-a$ to $a$, $a+ib$ to $-a+ib$, $\int_{R} e^{-\zeta^2}d\zeta=0$, however, the integral consists of four parts: + +Path 1: $-a\to a$ + +$\int_{I_1}e^{-\zeta^2}d\zeta=\int_{-a}^{a}e^{-\zeta^2}d\zeta=\int_{-a}^{a}e^{-x^2}dx$ + +Path 2: $a+ib\to -a+ib$ + +$\int_{I_2}e^{-\zeta^2}d\zeta=\int_{a+ib}^{-a+ib}e^{-\zeta^2}d\zeta=\int_{0}^{b}e^{-(a+iy)^2}dy$ + +Path 3: $-a+ib\to -a-ib$ + +$-\int_{I_3}e^{-\zeta^2}d\zeta=-\int_{-a+ib}^{-a-ib}e^{-\zeta^2}d\zeta=-\int_{a}^{-a}e^{-(x-ib)^2}dx$ + +Path 4: $-a-ib\to a-ib$ + +$-\int_{I_4}e^{-\zeta^2}d\zeta=-\int_{-a-ib}^{a-ib}e^{-\zeta^2}d\zeta=-\int_{b}^{0}e^{-(-a+iy)^2}dy$ + +> #### The reverse of a curve 6.9 +> +> If $\gamma:[a,b]\to\mathbb{C}$ is a curve, then the rever of $\gamma$ is the curve $-\gamma:[-b,-a]\to\mathbb{C}$ defined by $(-\gamma)(t)=\gamma(a+b-t)$. It is the curve one obtains from $\gamma$ by traversing it in the opposite direction. +> +> - If $\gamma$ is piecewise in $C^1$, then $-\gamma$ is piecewise in $C^1$. +> - $\int_{-\gamma}f(z)dz=-\int_{\gamma}f(z)dz$ for any function $f$ that is continuous on $\gamma([a,b])$. + +If we keep $b$ fixed, and let $a\to\infty$, then + +> #### Definition 6.10 (Estimate of the integral) +> +> Let $\gamma:[a,b]\to\mathbb{C}$ be a piecewise $C^1$ curve, and let $f:[a,b]\to\mathbb{C}$ be a continuous complex-valued function. Let $M$ be the maximum of $|f|$ on $\gamma([a,b])$. ($M=\max\{|f(t)|:t\in[a,b]\}$) +> +> Then +> +> $$\left|\int_{\gamma}f(\zeta)d\zeta\right|\leq L(\gamma)M$$ + +_Continue on previous example, we have:_ + +$$ +\left|\int_{\gamma}f(\zeta)d\zeta\right|\leq L(\gamma)\max_{\zeta\in\gamma}|f(\zeta)|\to 0 +$$ + +Since, + +$$ +\int_{-\infty}^{\infty}e^{-x^2}dx-\int_{-\infty}^{\infty}e^{-x^2+b^2}(\cos 2bx+i\sin 2bx)dx=0 +$$ + +Since $\sin 2bx$ is odd, and $\cos 2bx$ is even, we have + +$$ +\int_{-\infty}^{\infty}e^{-x^2}dx=\int_{-\infty}^{\infty}e^{-x^2+b^2}\cos 2bxdx=\sqrt{\pi}e^{-b^2} +$$ + +##### Proof for the last step: + +$$ +\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi} +$$ + +Proof: + +Let $J=\int_{-\infty}^{\infty}e^{-x^2}dx$ + +Then + +$$J^2=\int_{-\infty}^{\infty}e^{-x^2}dx\int_{-\infty}^{\infty}e^{-y^2}dy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dxdy$$ + +We can evaluate the integral on the right-hand side by converting to polar coordinates. $x=r\cos\theta$, $y=r\sin\theta,dxdy=rdrd\theta$ + +$$ +J^2=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dxdy=\int_{0}^{2\pi}\int_{0}^{\infty}e^{-r^2}rdrd\theta +$$ + +$$ +J^2=\int_{0}^{2\pi}\int_{0}^{\infty}e^{-r^2}rdrd\theta=\int_{0}^{2\pi}\left[-\frac{1}{2}e^{-r^2}\right]_{0}^{\infty}d\theta +$$ + +$$ +J^2=\int_{0}^{2\pi}\frac{1}{2}d\theta=\pi +$$ + +$$ +J=\sqrt{\pi} +$$ + +EOP + +## Chapter 7 Cauchy's theorem + +### Cauchy's theorem (Fundamental theorem of complex function theory) + +Let $\gamma$ be a closed curve in $\mathbb{C}$ and let $u$ be an open set _containing $\gamma^*$_. Let $f$ be a holomorphic function on $u$. Then + +$$ +\int_{\gamma}f(z)dz=0 +$$ + +Note: What "containing $\gamma^*$" means? (Rabbit hole for topologists) + +#### Lemma 7.1 (Goursat's lemma) + +Cauchy's theorem is true if $\gamma$ is a triangle. + +Proof: + +We plan to keep shrinking the triangle until $f(\zeta+h)=f(\zeta)+hf'(\zeta)+\mu(h)$ where $\mu(h)$ is a function of $h$ that goes to $0$ as $h\to 0$. + +Let's start with a triangle $T$ with vertices $z_1,z_2,z_3$. + +![carving a triangle](https://notenextra.trance-0.com/Math416/Cauchy_theorem_triangle_carving.png) + +We divide $T$ into four smaller triangles by drawing lines from the midpoints of the sides to the opposite vertices. + + + diff --git a/pages/Math416/_meta.js b/pages/Math416/_meta.js index c91bf35..c9e71fa 100644 --- a/pages/Math416/_meta.js +++ b/pages/Math416/_meta.js @@ -13,4 +13,5 @@ export default { Math416_L8: "Complex Variables (Lecture 8)", Math416_L9: "Complex Variables (Lecture 9)", Math416_L10: "Complex Variables (Lecture 10)", + Math416_L11: "Complex Variables (Lecture 11)", } diff --git a/public/Math416/Cauchy_theorem_triangle_carving.png b/public/Math416/Cauchy_theorem_triangle_carving.png new file mode 100644 index 0000000000000000000000000000000000000000..d41e3d7598d5c372305d9874d84164351ef540c1 GIT binary patch literal 83474 zcmeFZhdb4M{6BsuDpJU(6p};;SuHw7g^0+^PWH^ojLK?iD0`Kiy&aAUY1n&<>evSv zhYtBYUx)kq`TPyP>vLb%eRbciob!IaU(fM)J|2(fJ51y1Wd^!~bQlbVL0L&r6N8~) z!CZUH80C!a+(fJ!2fKwl2enzVDg{QubI)n|L<~E(s#vR_SB$% zsp_55Eijmk6lF!Z>vxQ&2Y0+;(Z-dGB=8>C{*{~g*rSK{0_ga8?mdjuxOXo-l1XuQ zB2T0!hL1(@q9}bN_SHMyZOX4=-zq(Q5wxCbMedA^GTK*RmFjbLpxa>f>~MaTk9enc zr;+;X+{SMBZuHOg@s08gs?Gn#Xv!z<-26{&)9nwpen@@k|9*(U{NDqigv0*@19}Yq zzYB)8BGZr53(NgZF+ud~6?>^d#kbO*`{L0{k41lfevg8 zZe53%{qJuPgu-T7afX8ub2JY=UjO>ySYz0};{$JYFox`8p_IZZQ7{7sL23v0)&H*L z_vdF^PTtGYHcc`lLleeGefJ+9s7m!QE)TIx=o13?i~QHU7v}4^`pcOLt(uASnEM$E z%@>A?Y&*Y^hWD|}wVagvM0Q*`ZX$FlKy)jaIQ;x76Fq&!ottS@l1EGiV+f4A5ZC zZ=HZ5e|Fc}Cwh^{OHU3)t*tJQ@8Pjonkgz$j}37i*?x_`9`6_XL^_`<Z{u<`%w z&|gxnPQ_UN-?PR<*E{zW*`(?hTH-ZCB2U~XdG9q}$KKeMa%GTDv&eDq?C{hb_xi)0 zWii`AUAHdQf@w)HR_tbQ`B=QqDGBN1V1?m3Kc85x%#Aqiqf!>um~MR=f;$s%`tnxf zL>`v$?w}3RHFOCQ`9Pog?Tv_&vEP!mxK-1k(y+wJe}AX&-ODOwM=`ux-446ozPF%m zZEc~s^|$^?4ePkJmdhwHp)V?%VT6MJ-SF-t;+JF_UrDZTyGh9?aGaJ@_P2y=ILcdeB zq(fhknAe|Q^E7UOIuPLw;purI9TlFlM7!avV^?0ov-|IN4y2gX+~2XcL@trz8;w@B zLGdYwy^@-DH6uSh-cQxbH^<)%$%LD?Y)@4qYGM8hrNtJNMuWrpt{kZnJIuJXQ&kTv zOcypsvkK}&-mHIG-uR#+aN7>w4bI%Y5{D_cFSe~ql+(*I4J@BXjZvLwOYu8mn3&S} zau!c^To`a|?lo)EzbKbVc3QCg5I_YF&~z)P!h>&pf884qt$tzM?WMGnvB?UXHQZvy z;_UG7^dPR=f#=HkzvFN5ci-x|jy1VU@4!6Tig=6I(=MOh8x((Yo9=1NDVe5q4Ag-~ z-guGS$RhP>ob6Mt3q+}xx30#Yx7jcvR_k_FdbyHG+pxE{!cpnh+(?b(uP@IJ7?@t9 zvZx7WXoQq>+H?5qFDWJ_mi-kQj=3kR*KT{dVwA9(!4>K=nO*8WrNhCilWRQ4o=0O) ze*1TU+~~^u&p|$(rrNN5ejEO|#{R3f{`W80aot_ZpBN}F6uC~H`jB13@t~wF3#(l6 zW!EzmcSCd`nLMR;1Z>|7kASkbz6E3~iFHH4X83}2uaGx6B<1tnV zG{^g{>V?x_Gq-NUMIlR4P)SHHFHuB^zDVV@kj-Rwt`mNJ`3C9t+i&_-algs^*A3EX zFq#Tmw_&qmFQ-I;|KALo*5n`o&3B<9So$Is-sSPu8X*(>fCJB@{Xki3?Coy?`08s* zgczram~B|*tqY7Sa+}a}9rhzdxi-smvq#ftu^T!E9uT*DtewpGzX4+!e!6pbXvD+<3FrL_#y8Tk~{xK0{&A~eQW?KyP%ht$5Mhit-=#*o4kNpD zdFd(K*N&Ka>)I}7-Y9jnT>radx$$q^ai5q~NBT97AX`ta^v_QZqqzNlzfo|revkN_ zxgWcCgcaL%o*Kk?O2@cZ&(rR|wg&G(#W3_FlitISF?Lv%YKkJ{x6x{m$clmkd)wr{eE#X324@0rH(CXR5 zaCP<9gbURoi*%_)b4pMcHcRMe7ksfF`D<3|+sjZoumxKIqVEXNM1TTX_~`AsB5D_I zRNh?M)uf|5hrH??s57=d>>Cr*dG8EiXt9--1r5+!?ZYG6Lc5*Q-gLTVkOiZ;b;LWf_8q?(Nkh-p!+lNeHHRaO0UdKLj(Bzf(h8@> zeE{b)SPGzu>iu>Z$ndPlaL@T8xx~YQza`Upbw`212)Y)CxH)7rxdw*Sncy`>U6w(+Y zop##f)=NIM0k}!+t*_jD#$!6t$gCV1ZK`4UZTyE5EhmgBippn7`lopWz11w@=YQ6d zmrHdIUfP5C2v>@t|J@@UTb*#d0t;mmC05lqk3Jj1>R5s_&SU0w43=-AJ2%0pI_ox} z!6sEL20faLtw*!n9#9-1h}}3(h6(EC41M~&jPECL4GocEHHS_+e{i6UdVlThCH#8s zHwYwjhhdv#H9QjqUxc7;G#%@SNO-DsUXG`z%zcc zLAt^>O57lFB(@^Qmt{HoMrqUHWaoDPfOYJOG+H!tETYdY{40Y?zdK;bNDIXyZdPV~ z5<=4TfBhyif)DSFa9wW1fnIi58m)_u*d(8DNK55b|EuUNcTMZx{VCmRfq*B~XTcxk z5aQ8WA#OQ4RAsq1-b(1Y;j|QV_2cz0=3}bxvqpF>5V16SxMX8T*AQw&Ul_2N!r0ip zQm2H~pU*BdAbdpuZ$DV91tK#|s`0D8UaC)v+9*CXxx3y;^7ra2akv3hediXutc_YH zlltdop&F5bR6oLySlxd)}0>&en^em8bsLD_sp`2vEX=eaV;_j2zIe6nC?eNY5 zk(y4m@iIQBv6eM*e;T;Ev5qG-Z}o62cJt1-gN30YpI`ZLKh?O(8;=!;Y~vAx=PL!R zSQ5&(Ta4WHws4Q2YNF3XXOcq5KpN)%{VJ{B`ck)tk}Kd|KRHd&6zn8I!&y4n5IYar z$pn0Yu{-om)JOg3Fxww&Mu2%Oq*()0l9_PAp)PBze(|$BcP{L-CSc2y#kIyG+ORcU6NdWi_d#2y$bucCqLHdZ;Fx5 zot0U(9czjMyz$>`#TeA#I+tJ1DSq)Gix`xTMVd>2T~Su?Htc%()&4TBM?Z(d+(90) z%WXP4B?P5lqfDJDAYe1xu&+*g{dsrBcO?&1MVpqyKoNerA}F4g1y;?D6>byeQVxB; zr0(wzH%PqGZRCl9Mq}&IemV@eUN!1e-T6OXj^ds9TQo!Upfa{+XdP$M3)=>$8Sbze zO8)ks%}2x2^8D!Kh7yN<`|zIk`r;=1EUIz-er=_h~`$DHba-os)^C=pko4x$tM z?+PCYV7vh#t3fO|K#lvxzMaMZ{ygPWIUgK#&NHWf_vizib&qHCxsK((*-&^M+yrA5 zOu_}H?7D

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