Struggling with Logarithmic Expansion - Can You Help?

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Discussion Overview

The discussion revolves around the expansion of the logarithmic function log[(x+epsilon)^2 + y^2] as epsilon approaches 0. Participants are seeking assistance with the mathematical expansion, exploring various approaches and clarifications related to logarithmic properties and series expansions.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant expresses difficulty in expanding log[(x+epsilon)^2 + y^2] as epsilon approaches 0 and requests help.
  • Another participant suggests expanding around x^2 + y^2 and provides a formula involving logarithmic properties.
  • A participant proposes using the series expansion log(1+a) = a - a^2/2 + ... for the first term but seeks clarification on how to handle log(x^2 + y^2).
  • It is noted that log(x^2 + y^2) should remain fixed and not be expanded.
  • A participant summarizes their overall expansion, incorporating the earlier suggestions and expressing gratitude for the help received.
  • Another participant provides a more detailed expansion, including higher-order terms and simplifications.

Areas of Agreement / Disagreement

Participants generally agree on the approach to expanding the logarithmic function, but there is some uncertainty regarding the treatment of log(x^2 + y^2) and the specifics of the expansion process.

Contextual Notes

Participants have not resolved the details of the expansion process, and there are assumptions about the behavior of terms as epsilon approaches 0 that remain unexamined.

Tranquillity
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Hello guys,

I am finding difficulties expanding log[(x+epsilon)^2 + y^2] as epsilon->0.

Could anyone help me with the expansion?

Kindest regards
 
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Tranquillity said:
Hello guys,

I am finding difficulties expanding log[(x+epsilon)^2 + y^2] as epsilon->0.

Could anyone help me with the expansion?

Kindest regards

Welcome to MHB, Tranquillity! :)

You would have to expand around $x^2+y^2$.
To do so, it's easiest if you use:

\[\log((x+\varepsilon)^2+y^2)=\log\left({x^2+y^2+2x \varepsilon +\varepsilon^2 \over x^2 + y^2}(x^2+y^2)\right) = \log\left(1 + {2x\varepsilon +\varepsilon^2 \over x^2 + y^2}\right) + \log(x^2+y^2)\]

Do you know how to expand that?
 
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ILikeSerena said:
Welcome to MHB, Tranquillity! :)

You would have to expand around $x^2+y^2$.
To do so, it's easiest if you use:

$\log((x+\varepsilon)^2+y^2)=\log({x^2+y^2+2x \varepsilon +\varepsilon^2 \over x^2 + y^2}(x^2+y^2)) = \log(1 + {2x\varepsilon +\varepsilon^2 \over x^2 + y^2}) + \log(x^2+y^2)$

Do you know how to expand that?

I could use that log(1+a) = a-a^2/2+... for the first parenthesis right?

But then how should I expand log(x^2+y^2)?

Thanks for that!
 
Tranquillity said:
I could use that log(1+a) = a-a^2/2+... for the first parenthesis right?

But then how should I expand log(x^2+y^2)?

Thanks for that!

Right!

You're not supposed to expand $\log(x^2+y^2)$.
It's supposed to be fixed.
 
ILikeSerena said:
Right!

You're not supposed to expand $\log(x^2+y^2)$.
It's supposed to be fixed.

So my overall expansion should be [(2*x*epsilon + epsilon^2) / (x^2+y^2) ] + log(x^2+y^2) ?

Thanks for all the help, it is greatly appreciated :)
 
Tranquillity said:
So my overall expansion should be [(2*x*epsilon + epsilon^2) / (x^2+y^2) ] + log(x^2+y^2) ?

Thanks for all the help, it is greatly appreciated :)

In an expansion you get $\displaystyle\left[{2*x*\varepsilon + \varepsilon^2 \over x^2+y^2} - \frac 1 2 \left({2*x*\varepsilon + \varepsilon^2 \over (x^2+y^2)}\right)^2 + ...\right.$

At first order this simplifies to:

$\displaystyle{2*x*\varepsilon \over x^2+y^2} + ...$
 
Last edited by a moderator:
Thanks for all the help!

Kindest regards
 

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