Intereting Posts

The most complete reference for identities and special values for polylogarithm and polygamma functions
Is this urn puzzle solvable?
Connection between Hermite & Legendre polynomials
Over $\mathbb{R}$, if $Z(p') \subset Z(p)$ when does $p' \vert p$?
Euclidean distance proof
Reflection principle for simple random walk
Without using prime factorization, find a prime factor of $\frac{(3^{41} -1)}{2}$
Representations of integers by a binary quadratic form
Convolution of two Gaussians is a Gaussian
Example where union of increasing sigma algebras is not a sigma algebra
Strehl identity for the sum of cubes of binomial coefficients
Learning Complex Geometry – Textbook Recommendation Request
Find all integer solutions of $1+x+x^2+x^3=y^2$
When can you treat a limit like an equation?
Are Monotone functions Borel Measurable?

Recall the definitions of the sine and cosine integrals:

$$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$

Both functions are oscillating, with a countably infinite number of minima and maxima.

Note that

$$\lim_{x\to\infty}\operatorname{Si}(x)=\frac\pi2,\quad\lim_{x\to\infty}\operatorname{Ci}(x)=0.$$

Consider the following function:

$$f(x)=\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}.$$

It appears that the function $f(x)$ and all its derivatives are monotonic for $x>0$. Specifically, the function itself and all its derivatives of an even order are strictly decreasing, and all its derivative of an odd order are strictly increasing.

Is it actually true? If so, then how can we prove it?

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- Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$
- Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?
- How should I calculate the $n$th derivative of $f(x)=x^x$?
- Resources for learning Elliptic Integrals

- Probability that a random permutation has no fixed point among the first $k$ elements
- A closed form of the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$
- Infinite product: $\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}$
- Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge?
- Some pecular fractional integrals/derivatives of the natural logarithm
- Difficult Integral: $\int\frac{x^n}{\sqrt{1+x^2}}dx$
- How to integrate $\int \frac{e^x dx}{1\,+\,e^{2x}}$
- Nth derivative of a function: I don't know where to start
- Computing $\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}+1\right)}{k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\right)}$ in closed form
- Interpretation of $\epsilon$-$\delta$ limit definition

- Change of limits in definite integration
- Find the number of arrangements of $k \mbox{ }1'$s, $k \mbox{ }2'$s, $\cdots, k \mbox{ }n'$s – total $kn$ cards.
- Help on differential equation $y''-2\sin y'+3y=\cos x$
- Can someone intuitively explain what the convolution integral is?
- Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris
- Axioms for sets of numbers
- Find $\frac{\mathrm d^{100}}{\mathrm d x^{100}}\frac{x^2+1}{x^3-x}=$?
- Algebraic closure for rings
- Same number of partitions of a certain type?
- Deciding if a number is a square in $\Bbb Z/n\Bbb Z$
- The limit of the derivative of an increasing and bounded function is always $0$?
- Showing $ \times $ is a manifold with boundary
- Real Analysis Boundedness of continuous function
- Prove that $a+b$ is a perfect square
- What does the dot product of two vectors represent?