Hi I am just wondering,
If I have a signal $f[n]\in \mathbb{C}^L$, i.e. $f$ is $L$-periodic, i can also define $h[n]=f[-n]$.
Is it true that the Fourier transform of $f$, say $\hat{F}$, and the Fourier tranform of $h$, say $\hat{H}$ is related by the equation below?
$$|\hat{F}[k]|=|\hat{H}[k]|...
1
Here is an elementary way to evaluate the integral without involving any special functions or advance formulas. Notice that
$$\int_0^\infty e^{-y}\sin(xy)\;\mathrm dy=\frac{x}{1+x^2}$$
Hence we have
\begin{align}
\int_0^\infty \frac{x}{\left(1+x^2\right)\left(e^{2\pi x}+1\right)}\,\mathrm dx&=\in...
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
It is the first of the polygamma functions.
== Relation to harmonic numbers ==
The digamma function, often denoted also as ψ0(x), ψ0(x) or (after the shape of the archaic Greek letter Ϝ digamma), is related to the harmonic numbers in that
where Hn is the n-th harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as
== Integral representations ==
If the real part of x is positive then the digamma function has the following integral representation...
