Term-by-term integration and differentiation of Fourier series

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Problem 11A.10 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11A "Fourier Series and Poisson Integral"

Suppose $f: \partial \mathbf{D} \rightarrow \mathbb {C}$ is three times continuously differentiable. Prove that

f^{[1]}(z) = i \sum _{n = -\infty }^{\infty } n \widehat{f}(n) z^{n}

for all $z \in \partial \mathbf{D}$ .

Problem 2962 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.6 "Fourier Series"

From the expansion

x=2\sum _{n=1}^{\infty } (-1)^{n+1}\frac{\sin nx}{n} \quad (-\pi <x<\pi ),

using term-by-term integration, find the Fourier series of the functions $x^{2}$ , $x^{3}$ , and $x^{4}$ on the interval $(-\pi ,\pi )$ .

Problem 2973 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 5.6 "Fourier Series"

Expand the function

f(x)=\int _{0}^{x} \ln \sqrt{\left|\operatorname {ctg}\frac{t}{2}\right|}\, dt \quad (-\pi \leqslant x \leqslant \pi )

into a Fourier series.

Problem 4852 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 11.8 "Miscellaneous Problems"

Using the identity

4\cos ^{2}x = (1+e^{2xi})(1+e^{-2xi}),

prove the equalities, for $0 < x < \dfrac {\pi }{2}$ ,

\int _{0}^{x} \ln \cos x\, dx = -x\ln 2-\frac{1}{2}\sum _{\nu =1}^{\infty } \frac{(-1)^{\nu }}{\nu ^{2}}\sin 2\nu x,

\int _{0}^{x} \ln \sin x\, dx = -x\ln 2-\frac{1}{2}\sum _{\nu =1}^{\infty } \frac{1}{\nu ^{2}}\sin 2\nu x .

(N.I. Lobachevsky.)

Problem 2.22.62 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

(1)(2)

Using the expansion of the function $f(x) = x$ into a Fourier series (see problem 4), prove that:

(1)

$x \sin x = 1-\frac{1}{2} \cos x-2 \sum_{n = 2}^{\infty } \frac{(-1)^{n} \cos n x}{n^{2}-1},-\pi \leqslant x \leqslant \pi$ ;

(2)

$x \cos x = -\frac{1}{2} \sin x+2 \sum_{n = 2}^{\infty } \frac{(-1)^{n} n \sin n x}{n^{2}-1},-\pi < x < \pi$ .

Problem 2.22.112 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Expand the function into a Fourier series

f(x) = \int _{0}^{x} \ln \sqrt{\left|\operatorname {ctg} \frac{t}{2}\right|} d t, \quad -\pi \leqslant x \leqslant \pi

Problem 2.22.116 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

(1)(2)(3)

Starting from the expansion

x = 2 \sum _{n = 1}^{\infty }(-1)^{n+1} \frac{\sin n x}{n}, \quad -\pi < x < \pi

obtain, by term-by-term integration, the Fourier series expansion of the function:

(1)

$x^{2}$ ;

(2)

$x^{3}$ ;

(3)

$x^{4}$ .

Problem 2.22.120 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Интегралы. Ряды." [ККЧ+03]More problems in Subsection 2.22.2 "Problems"

Prove that if the function $f$ is absolutely integrable on the segment $[-\pi ; \pi ]$ ,

a_{0}+\sum _{n = 1}^{\infty }\left(a_{n} \cos n x+b_{n} \sin n x\right)

is its Fourier series, then for any points $x^{\prime } \in [-\pi ; \pi ]$ and $x^{\prime \prime } \in [-\pi ; \pi ]$ the equality holds

\int _{x^{\prime }}^{x^{\prime \prime }} f(x) d x = \int _{x^{\prime }}^{x^{\prime \prime }} a_{0} d x+\sum _{n = 1}^{\infty } \int _{x^{\prime }}^{x^{\prime \prime }}\left(a_{n} \cos n x+b_{n} \sin n x\right) d x.

References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Интегралы. Ряды. 2-е изд., ФИЗМАТЛИТ, 2003.