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diff --git a/kang/hw7.tex b/kang/hw7.tex new file mode 100644 index 0000000..6ca5f51 --- /dev/null +++ b/kang/hw7.tex @@ -0,0 +1,92 @@ +\def\Re{\mathop{\rm Re}\nolimits} +\def\Im{\mathop{\rm Im}\nolimits} +\def\Log{\mathop{\rm Log}\nolimits} +\let\rule\hrule +\def\hrule{\medskip\rule\medskip} + +% page 147 +\noindent{\bf 2.} + +\noindent{\it (a)} + +The antiderivative of $z^2$ is $z^3/3$ (because the derivative of that +function is $z^2$) + +$$\int_0^{1+i} z^2 dz = \left[ {z^3\over3} \right]_0^{1+i} = (2/3)(1-i).$$ + +\noindent{\it (b)} + +$2\sin(z/2)$ differentiates to $\cos(z/2),$ so it is the antiderivative. +The integral, then, evaluates to: + +$$2(\sin(\pi+2i) - \sin 0) = $$ + +\noindent{\it (c)} + +The antiderivative of the integrand is $(z-2)^4/4,$ so the integral +evaluates to +$$\left[ (z-2)^4/4 \right]_1^3 = ( (3-2)^4 - (1-2)^4 )/4 = 0.$$ + +\hrule +%page 159 +\noindent{\bf 2.} + +The corollary tells us that these two integrals will be the same if the +functions are analytic on $C_1,$ $C_2,$ and the region between them. + +\noindent{\it (a)} + +With $$f(z) = {1\over 3z^2 + 1},$$ +$f = (3z^2 + 1)^{-1} = f_1(f_2)$ can be differentiated with the chain +rule if $f_1'$ and $f_2'$ are defined. $z^{-1} = f_1$ is analytic where +$f_2 \neq 0,$ and $3z^2 + 1$ is a polynomial, so it is entire. +Because $3z^2 + 1 = 0 \to |3z^2| = 1 \to |z| = \sqrt{1/3}$ is not within +the described region (it is excluded by the square of radius 1), the +equality holds. + +\noindent{\it (b)} + +Similarly, $$f(z) = {z+2\over \sin(z/2)}$$ +implies the equality holds if $z+2$ and $\sin(z/2)$ are analytic and +$\sin(z/2) \neq 0.$ These are both entire functions, so the first +criterion is trivial. +From an earlier theorem, we know that the only zeroes of $\sin z$ are $z += n\pi$ with $n\in\{0,\pm 1,\pm 2,\ldots\}$ Therefore, $\sin(z/2) = 0$ +requires $z = 2n\pi.$ +However, none of these points are included in the given region because, +where $n\neq0,$ $|z| > 4,$ so they are outside of the bounding circle, +and when $n=0,$ $z=0$ is excluded by the square. + +% i would like to remember this theorem better +% and state the proof more nicely + +\noindent{\it (c)} + +Again, $$f(z) = {z\over 1-e^z}$$ +requires $1 - e^z = 0 \to e^z = 1 \to z = \log 1 = 2n\pi i$ (with $n$ +constrained to the integers). +These points are not contained within the circle $|z|=4,$ except $z=0,$ +which is excluded by the square of radius 1. + +\noindent{\bf 6.} + +The Cauchy-Goursat theorem doesn't apply here because this requires a +branch cut at $\theta = -{\pi\over2},$ so $f(z) = \sqrt{r}e^{i\theta/2}$ +isn't continuously defined on the region. +On the semicircle, $r = 1,$ $\pi \in (0, \pi),$ and $z' = ie^{i\theta},$ +so $$\int_0^\pi (\sqrt{1}e^{i\theta/2})(ie^{i\theta}) d\theta += \int_0^\pi ie^{3i\theta/2} d\theta += \left[ (2/3)e^{3i\theta/2} \right]_0^\pi += (2/3)(e^{3\pi i/2} - 1) = (2/3)(-i-1).$$ + +On the two radii $\theta = 0,\pi,$ $r\in (0,1),$ the integral is +evaluated as +$$\int_0^1 \sqrt{r}e^{i\theta/2}(ie^{i\theta}) dr += e^{3i\theta/2} \int_0^1 \sqrt{r} dr += e^{3i\theta/2}(2/3).$$ +This is $-(2/3)i$ where $\theta = \pi,$ (evaluated in the negative +direction, so it is $(2/3)i$ in the positive direction) and $2/3$ where +$\theta = 0.$ +Summing these results, we get $0.$ + +\bye |