complex analysis - tricky question on poles.

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The Racist Dragon
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#1
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Let f and g be functions with a pole at c. Create rules (and prove them) about how we can combine f and g at c.

and ii: Find the poles of the function :
\frac{cotz+cosz}{sin2z}

and classify these poles using part i.

I have attempted this, but I keep getting poles of negative order in part ii and my course apparently doesn't focus on poles of negative order so I must have gone wrong somewhere!
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davros
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(Original post by The Racist Dragon)
Let f and g be functions with a pole at c. Create rules (and prove them) about how we can combine f and g at c.

and ii: Find the poles of the function :
\frac{cotz+cosz}{sin2z}

and classify these poles using part i.

I have attempted this, but I keep getting poles of negative order in part ii and my course apparently doesn't focus on poles of negative order so I must have gone wrong somewhere!
Have you started by rewriting cot z = cos z / sin z and sin2z = 2sinzcosz and simplifying?
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The Racist Dragon
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(Original post by davros)
Have you started by rewriting cot z = cos z / sin z and sin2z = 2sinzcosz and simplifying?
*sigh* that's probably where I went wrong, I simplified most of it but forgot that sin2z=2sinzcosz, but is it legal to expand sin2z too? There was nothing wrong with the taylor series for sin2z..!
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davros
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(Original post by The Racist Dragon)
*sigh* that's probably where I went wrong, I simplified most of it but forgot that sin2z=2sinzcosz, but is it legal to expand sin2z too? There was nothing wrong with the taylor series for sin2z..!
Oh I'm sure it's "legal" to expand sin2z - I just tend to rewrite things in terms of "basic" functions that make things (hopefully) more transparent!
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Smaug123
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(Original post by The Racist Dragon)
*sigh* that's probably where I went wrong, I simplified most of it but forgot that sin2z=2sinzcosz, but is it legal to expand sin2z too? There was nothing wrong with the taylor series for sin2z..!
Yes, that identity holds even for complex z. Cancelling a sin from top and bottom is just like using x instead of x^2/x - it's just as legal as that.
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