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I'm relatively new here( This is my first question tbh) This question is from my assignment which I'm supposed to submit in two days. I have been scratching my head over past couple of days but have reached nowhere.

Obtain the first three terms in the expansion of function F(x) in a series of the form $$ F(x) = \sum_{k=0}^{\infty} A_k P_k(x) $$ where $$F(x)=\{\cos(x) \text{ for } 0 \le x \le \pi/2 \ $$$$0 \text{ for } \frac{\pi}{2} \le x \le \pi.\} $$

What I know is I have to use legendre's expansion formula i.e,$F(x) =\sum A_kP_k(x)$ where $-1≤x≤1$ But obviously I cannot use it directly because the range of $x$ differs. I have tried substituting $x=\cos(\theta)$ but no success so far.

P.S Please don't downvote, and instead suggest where I must improve. Thank you

Yash Chugh
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  • Welcome to Math.SE, I hope you stay to contribute to the site! Here is a MathJax tutorial: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – gt6989b Sep 12 '17 at 15:43
  • Thank you very much for your invaluable help! Will stay and contribute surely – Yash Chugh Sep 12 '17 at 15:58
  • I do not understand your definition of $F(x):$ Is it $\cos x$ on the whole interval $[0, \pi]$ or is there something mission before 'or for ...'.? You can use an affine ('linear') transformation, $u=a + bx$. – gammatester Sep 12 '17 at 16:20
  • I am extremely sorry for that. The last edit messed it and didn't notice! – Yash Chugh Sep 12 '17 at 16:38
  • Hi @gammatester, I have, with much pain, corrected my previous mistakes. Thank you. Do point if there is anythingr else wrong or missing. – Yash Chugh Sep 12 '17 at 16:52
  • The differing ranges $0\le x\le \pi$ and $-1\le x\le 1$ is troubling. Despite this, $P_n(x)$ is a polynomial in $x$ and is defined for all $x$. The trouble is that to use the orthogonality of $P_n(x)$ you need to integrate on the interval $-1\le x\le 1$. Please double check your question source to determine the source of the discrepancy. – Somos Sep 12 '17 at 17:25
  • I did. It's the same. And it's the range that is bugging me too – Yash Chugh Sep 12 '17 at 19:24

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