MHB Struggling with Logarithmic Expansion - Can You Help?

[Tranquillity]

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

 

[I like Serena]

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?

 

[Tranquillity]

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!

 

[I like Serena]

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.

 

[Tranquillity]

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 :)

 

[I like Serena]

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:

$\displaystyle{2*x*\varepsilon \over x^2+y^2} + ...$

 

[Tranquillity]

Thanks for all the help!

Kindest regards