- #1

Lambda96

- 189

- 65

- Homework Statement
- ##\bigl\langle f,\cal L g \bigr\rangle=\bigl\langle \cal L f,\ g \bigr\rangle##

- Relevant Equations
- none

Hi,

unfortunately, I have problems with the following task

I tried the fast way, unfortunately I have problems with it

I have already proved the following properties, ##\bigl< f,xg \bigr>=\bigl< xf,g \bigr>## and ##\bigl< f, \frac{d}{dx}g \bigr>=-\overline{f(0)} g(0)+\bigl< f,g \bigr>-\bigl< \frac{d}{dx}f,g \bigr>## and then proceeded as follows:

$$\bigl< f,\cal L g \bigr>$$

$$\bigl< f,(-x\frac{d^2}{dx^2}+(x-1)\frac{d}{dx})g \bigr>$$

$$\bigl< f,-x \frac{d^2}{dx^2}g+x \frac{d}{dx}g-\frac{d}{dx}g \bigr>$$

$$\bigl< f,-x \frac{d^2}{dx^2}g \bigr>+\bigl< f,x \frac{d}{dx}g \bigr>-\bigl< f,\frac{d}{dx}g \bigr>$$

Then for the last term, I used the above identity ##\bigl< f,\frac{d}{dx}g \bigr>=-\overline{f(0)} g(0)+\bigl< f,g \bigr>-\bigl< \frac{d}{dx}f,g \bigr>## and obtained the following:

$$\bigl< f,-x \frac{d^2}{dx^2}g \bigr>+\bigl< f,x \frac{d}{dx}g \bigr>+\overline{f(0)} g(0)- \bigl< f,g \bigr>+\bigl< \frac{d}{dx}f,g \bigr>$$

Unfortunately now I'm stuck because I don't know what to do with the first two terms, in the hint it says to apply the above identities to ##\bigl< f,-x \frac{d^2}{dx^2}g \bigr>## but unfortunately I don't know how.

unfortunately, I have problems with the following task

I tried the fast way, unfortunately I have problems with it

I have already proved the following properties, ##\bigl< f,xg \bigr>=\bigl< xf,g \bigr>## and ##\bigl< f, \frac{d}{dx}g \bigr>=-\overline{f(0)} g(0)+\bigl< f,g \bigr>-\bigl< \frac{d}{dx}f,g \bigr>## and then proceeded as follows:

$$\bigl< f,\cal L g \bigr>$$

$$\bigl< f,(-x\frac{d^2}{dx^2}+(x-1)\frac{d}{dx})g \bigr>$$

$$\bigl< f,-x \frac{d^2}{dx^2}g+x \frac{d}{dx}g-\frac{d}{dx}g \bigr>$$

$$\bigl< f,-x \frac{d^2}{dx^2}g \bigr>+\bigl< f,x \frac{d}{dx}g \bigr>-\bigl< f,\frac{d}{dx}g \bigr>$$

Then for the last term, I used the above identity ##\bigl< f,\frac{d}{dx}g \bigr>=-\overline{f(0)} g(0)+\bigl< f,g \bigr>-\bigl< \frac{d}{dx}f,g \bigr>## and obtained the following:

$$\bigl< f,-x \frac{d^2}{dx^2}g \bigr>+\bigl< f,x \frac{d}{dx}g \bigr>+\overline{f(0)} g(0)- \bigl< f,g \bigr>+\bigl< \frac{d}{dx}f,g \bigr>$$

Unfortunately now I'm stuck because I don't know what to do with the first two terms, in the hint it says to apply the above identities to ##\bigl< f,-x \frac{d^2}{dx^2}g \bigr>## but unfortunately I don't know how.