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by Tolaso J Kos
Tue Jun 09, 2020 11:33 am
Forum: Functional Analysis
Topic: Inner product space
Replies: 1
Views: 338

Re: Inner product space

Hint: Equality holds when vectors are parallel i.e, $u=kv$, $k \in \mathbb{R}^+$ because $u \cdot v= \|u \| \cdot \|v\| \cos \theta$ when $\cos \theta=1$, the equality of the Cauchy-Schwarz inequality holds.
by Tolaso J Kos
Sun Dec 15, 2019 10:51 pm
Forum: Calculus
Topic: Digamma and Trigamma series
Replies: 0
Views: 2104

Digamma and Trigamma series

Let $\psi^{(0)}$ and $\psi^{(1)}$ denote the digamma and trigamma functions respectively. Prove that:

\[\sum_{n=1}^{\infty} \left ( \psi^{(0)}(n) - \ln n + \frac{1}{2} \psi^{(1)}(n) \right ) = 1+ \frac{\gamma}{2} - \frac{\ln 2\pi}{2}\]

where $\gamma$ denotes the Euler – Mascheroni constant.
by Tolaso J Kos
Sun Oct 13, 2019 1:06 pm
Forum: Archives
Topic: Mathematical newspaper
Replies: 1
Views: 1525

Re: Mathematical newspaper

The second issue of the JoM Journal is now out. You may download it from this web address. Hope you find something interesting within its $97$ pages.
by Tolaso J Kos
Sat Oct 12, 2019 12:26 pm
Forum: Blog Discussion
Topic: A logarithmic Poisson integral
Replies: 1
Views: 1330

A logarithmic Poisson integral

A logarithmic Poisson integral by Tolaso J Kos Let $a \geq 0$. We will prove that $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, \mathrm{d}x = \left\{\begin{matrix} 0 & , & \left | a \right | \leq 1 \\ 2 \pi \ln \left | a \right | &, & \text{otherwise} \end{matrix}\right.$$ Background: This...
by Tolaso J Kos
Thu Oct 03, 2019 9:38 am
Forum: Meta
Topic: Welcome to the new and improved mathimatikoi.org
Replies: 7
Views: 4618

Re: Welcome to the new and improved mathimatikoi.org

As of today we have the ability to include xy.pic into our posts. Unfortunately, the rendering of all equations takes a little time to complete. We'll see if we can overcome this problem.
by Tolaso J Kos
Thu Oct 03, 2019 9:04 am
Forum: LaTeX code testings
Topic: xy.jax
Replies: 4
Views: 1767

Re: xy.jax

$$\xymatrix{ {} & {} & {} & P\ar[d]^h\ar@{-->}[ldd] & {} \\ 0 \ar[r] & \mathrm{Ker}g\ar[r]^j & A\ar[r]^g\ar@{-->}[d]_i & B\ar[r]\ar@{-->}[d]^{\theta} & 0 \\ {} & {} & P\ar@{~>}[r]_-{k} & P/\mathrm{Im}(i\circ j) & {}
}$$
by Tolaso J Kos
Thu Oct 03, 2019 9:03 am
Forum: LaTeX code testings
Topic: xy.jax
Replies: 4
Views: 1767

Re: xy.jax

$$\xymatrix{& {P_{1}\oplus P_{2}}\ar@{->}[dldd]|{\boxed{\pi_{1}}}\ar[drdd]^{\pi_{2}} & \\
& & \\
& M\ar[dl]_{k_{1}}\ar[dr]^{k_{2}}\ar@{=>}[dd]_f^g\ar@{-->}[uu]_{\theta} & \\
P_{1}\ar[dr]_{h_{1}} & & P_{2}\ar[dl]^{h_{2}} \\
{}& N & {}
}$$
by Tolaso J Kos
Thu Oct 03, 2019 9:02 am
Forum: LaTeX code testings
Topic: xy.jax
Replies: 4
Views: 1767

Re: xy.jax

$$\xymatrix{ && &j\ar@/_/@{->}[ddlll]\ar@/^/@{--}[ddddd] &a\ar@/^/[l]\ar@/_/@{--}[ddddd] &&& \\ \\y \ar@/_/[d]&&&&& &
&u\ar@/_/[uulll]\\x\ar@/_/[ddrrr]& &&&&& & v\ar@/_/ \\\\ &&& b\ar@/^/[r] & i \ar@/_/[uurrr]&&& }$$
by Tolaso J Kos
Thu Oct 03, 2019 9:01 am
Forum: LaTeX code testings
Topic: xy.jax
Replies: 4
Views: 1767

xy.jax

$$\xymatrix{ U \ar@/_/[ddr]_y \ar@/^/[drr]^x \ar@{.>}[dr]|-{(x,y)} \\ & X \times_Z Y \ar[d]^q \ar[r]_p & X \ar[d]_f \\ & Y \ar[r]^g & Z }$$ $$\xymatrix{\boxed{\text{cases}}\ar@/^/[dr]!U|1\ar@/^/[drr]!U|2\ar@/^/[drrr]!U|3\ar@/^/[drrrr]!U|4\\&*+[F]\txt{x}&*+[F]\txt{x\\x}&*+[F-,]\txt{x\\x\\x}&*+ [F-:<8...
by Tolaso J Kos
Tue Oct 01, 2019 12:35 pm
Forum: Blog Discussion
Topic: Pythagoras' theorem 3D version
Replies: 0
Views: 1633

Pythagoras' theorem 3D version

Pythagoras' theorem 3D version by Tolaso J Kos The one theorem that almost anyone who has ever studied Mathematics remembers is the Pythagoras’ theorem. It is the most well-known theorem for about 2,500 years now and has allowed the mankind to evolve in a myriad of ways. Its applications are limitl...