By expanding in a power series in alpha, find the Fourier
By expanding in a power series in $\alpha$, find the Fourier series (in $x$) of $f(x)=\dfrac{\alpha\sin x}{1-2\alpha\cos x+\alpha^2}$, $|\alpha|<1$.
Explanation of Method
The denominator factors as $(1-\alpha e^{ix})(1-\alpha e^{-ix})$, so $f(x)$ is the imaginary part of the complex geometric series $\dfrac{\alpha e^{ix}}{1-\alpha e^{ix}}=\sum_{n\ge1}\alpha^ne^{inx}$. Taking imaginary parts directly gives a pure sine series with geometric coefficients.
- Identify $f(x)$ as $\operatorname{Im}\left[\dfrac{\alpha e^{ix}}{1-\alpha e^{ix}}\right]$.
- Expand the complex geometric series and take the imaginary part.
- Verify the rationalized real form matches $f(x)$ directly.
Step 1 · Complex identification
$$ \frac{\alpha e^{ix}}{1-\alpha e^{ix}} = \sum_{n=1}^\infty\alpha^ne^{inx} = \sum_{n=1}^\infty\alpha^n\cos(nx) + i\sum_{n=1}^\infty\alpha^n\sin(nx). $$
Step 2 · Rationalize and match
Multiplying numerator and denominator by $1-\alpha e^{-ix}$: $$ \frac{\alpha e^{ix}(1-\alpha e^{-ix})}{(1-\alpha e^{ix})(1-\alpha e^{-ix})} = \frac{\alpha e^{ix}-\alpha^2}{1-2\alpha\cos x+\alpha^2} = \frac{(\alpha\cos x-\alpha^2)+i\alpha\sin x}{1-2\alpha\cos x+\alpha^2}. $$ The imaginary part is exactly $f(x)$: $$ \operatorname{Im}\left[\frac{\alpha e^{ix}}{1-\alpha e^{ix}}\right] = \frac{\alpha\sin x}{1-2\alpha\cos x+\alpha^2} = f(x). $$
Step 3 · Fourier series
$$ \boxed{\frac{\alpha\sin x}{1-2\alpha\cos x+\alpha^2} = \sum_{n=1}^\infty\alpha^n\sin(nx)} $$ (Verified numerically at $\alpha=0.4,x=1.2$: $f(1.2)\approx0.42847$, series partial sum $\approx0.42847$, matching exactly.)
Summary Table
| Property | Value |
|---|---|
| Fourier series | $f(x)=\sum\alpha^n\sin(nx)$ |
| Coefficient | $b_n=\alpha^n$ |
Additional Notes
- This is the "sine Poisson kernel," the odd companion to the standard (cosine) Poisson kernel derived in Problem 347 — together they form the real and imaginary parts of $\alpha e^{ix}/(1-\alpha e^{ix})$.
- The same rationalization trick, splitting a single complex geometric series into a real (cosine) and imaginary (sine) Fourier series, is the same device used in Problem 262's $\sum x^n\cos(n\alpha)$/$\sin(n\alpha)$ pair from earlier in this collection.
- As $\alpha\to1^-$, the coefficients $\alpha^n\to1$ and this series approaches (non-uniformly) the conditionally-convergent series $\sum\sin(nx)$, related to the sawtooth-wave series of Problem 324.