Can someone explain to me how this is not 1/4? Complex Analy

[RJLiberator]

Homework Statement

Hi all, I posted the image of the solution here.

My questions concerns the evaluation of the Residue at z=i.

Homework Equations

None needed, all giving in the question -- simple algebraic mistake made is likely to be the problem here.

The Attempt at a Solution

[/B]

We see that phi function is clearly equal to 1/(z+i)^2 as the show.
Since it is a pole of order two, we must differentiate this function first before evaluation at singular point i.

The derivative of this function should be -2/(z+i)^(3)

Now we can evaluate this at point z=i.
-2/(2i)^3 = -2/8i^3 = -1/4i^3. When I calculate this, this becomes i/4

But the answer states that this is instead 1/4i.

Where am I going wrong in my complex algebra?

 

[QIsReluctant]

Homework Statement

Hi all, I posted the image of the solution here.

My questions concerns the evaluation of the Residue at z=i.

View attachment 86988

Homework Equations

None needed, all giving in the question -- simple algebraic mistake made is likely to be the problem here.

The Attempt at a Solution

[/B]

We see that phi function is clearly equal to 1/(z+i)^2 as the show.
Since it is a pole of order two, we must differentiate this function first before evaluation at singular point i.

The derivative of this function should be -2/(z+i)^(3)

Now we can evaluate this at point z=i.
-2/(2i)^3 = -2/8i^3 = 1/4i^3. When I calculate this, this becomes i/4

But the answer states that this is instead 1/4i.

Where am I going wrong in my complex algebra?

Looks like a simple algebra mistake. When you simplify for the second time you eliminate the negative sign, effectively dividing by -1. That's why your answer differs by a factor of -1.

 

[RJLiberator]

Wopps. good catch, that was a mistake in my post, but not the mistake i wanted to clarify. (I've seen edited my original post for future readers).

The problem is, when I evaluate this, it comes to -1/4i^3 and I believe this is equal to i/4 as per wolfram alpha, etc.
However, in the image, it shows that this is not the case. Instead, the image shows the solution to be 1/4i.

What's going on here? My only other line of thinking is this: They split up -1/4i^3 into -1/(4i^2*i)

Oh...

That works...

-1/(4i^2*i) = 1/4i
But, why would they not simplify this further so that it is i/4? The final answer would then be the negative of what the solution offers.

 

[QIsReluctant]

But, why would they not simplify this further so that it is i/4?
Because that would not be a valid simplification. 1/i = -i, and so we would need -i/4 and not i/4.

I'd be interested in seeing this Wolfram Alpha output you speak of ...

 

[RJLiberator]

Ah, I see. The that is indeed the mistake I was performing. Did not have parentheses around the denominator in wolfram alpha.

So it does come out to be -i/4. Which with that negative factor, makes them equivalent.

You have successfully helped me solve this problem and furthered my understanding.
Thank you.

 

[QIsReluctant]

Did not have parentheses around the denominator in wolfram alpha.

Aha! That's the reason.

Glad I could help.