#How to know contour shapes whether it's simple or not simple?

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ocean carbon
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I am studying complex analysis but having difficulty in understanding the shapes of contour loops. Example:
cos(z)/(z^2 -4) book writes integrand fails to be analytics at points z = +- 2, but one of them can occur inside. How to know which one? And as in the if diagram is not given, how am I supposed to know the loops and their orientation etc. Can someone provides an easy answer? Thanks

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ocean carbon
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ok I understood the book's question, the denominator is supposed to be in a form of z - z0 so makes sense to choose f(z) = cosz/(z+2)

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but I still not understand the question in the screen shot

timid fulcrum
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Basically, the orientation of a loop is chosen when you integrate, and depending on the contour you chose the contour may or may not contain poles where the function isn’t analytic

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Let’s give an example consider $f:z\rightarrow \frac{1}{z^{2}+1}$ for $R>1$ consider the set $C_{(a,b),R}={z \in \mathbb{C}, |z|=1, arg(z) \in [a;b]$ }$ $C_{(0,\pi),R}$ contains a pole of f which is i but not the other pole -i now if you consider $C_{(0,-\pi),R}$ it contains -i but not i

violet sapphireBOT
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ocean carbon
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can you briefly tell me what poles are? I forgot about it

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is it like cuts through real axis right?

timid fulcrum
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Basically if you have a function of the form 1/f then z is a pole of 1/f if f(z)=0 that’s the basic gist of it

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It’s the same idea as the poles for rational fractions

ocean carbon
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ok so basically what i said points through which f(z) cuts the Re axis of the plane

timid fulcrum
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Yes in your example z=2 and z=-2 are poles it’s where the function isn’t analytic in this case

ocean carbon
timid fulcrum
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the function here has poles i and -i do you agree ?

ocean carbon
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yes

timid fulcrum
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Well certain loops contain i or -i and others don’t

ocean carbon
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how do i know the loop look like

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by just with the equation

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in other words, how do I visualize the complex equation

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in this specific context

timid fulcrum
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Let $z_0$ be a complex number and $R>0$ now consider the set $C_R={z \in \mathbb{C}, |z-z_0|=R }$

violet sapphireBOT
timid fulcrum
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Here this is the equation of a circle so drawing it in the complex plane with give you a circle centered around z_0 and of radius R

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There are other examples like ellipses, etc you have to have an intuition on what is drawn

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The cool thing about contour integration is that the value of the integral only depends on which poles are contained within the contour ( using Cauchy’s integral theorem, and formula and the residue theorem)

ocean carbon
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oh ok understood

ocean carbon
low spearBOT
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