2023.12.12 闲话

双十二 .

HE NOIP . HE NOIP . HE NOIP . HE NOIP . HE NOIP . HE NOIP . HE NOIP . HE NOIP . HE NOIP . HE NOIP .

nzhtl1477

唉 lxl 你真是越来越没有意义了 .

跑到粉兔的粉丝群，好像回来看 lxl 倒是比较文艺了 .

歌：Sublimate Progression - MidLuster feat. 歌愛ユキ + 鏡音リン .

Welcome To The Internet - Bo Burnham

Welcome to the internet
Have a look around
Anything that brain of yours can think of can be found
We've got mountains of content
Some better, some worse
If none of it's of interest to you, you'd be the first

Welcome to the internet
Come and take a seat
Would you like to see the news or any famous women's feet?
There's no need to panic
This isn't a test, haha
Just nod or shake your head and we'll do the rest

Welcome to the internet
What would you prefer?
Would you like to fight for civil rights or tweet a racial slur?
Be happy
Be horny
Be bursting with rage
We got a million different ways to engage

Welcome to the internet
Put your cares aside
Here's a tip for straining pasta
Here's a nine-year-old who died
We got movies, and doctors, and fantasy sports
And a bunch of colored pencil drawings
Of all the different characters in Harry Potter fucking each other

Welcome to the internet
Hold on to your socks
'Cause a random guy just kindly sent you photos of his cock
They are grainy and off-putting
He just sent you more
Don't act surprised, you know you like it, you whore

See a man beheaded
Get offended, see a shrink
Show us pictures of your children
Tell us every thought you think
Start a rumor, buy a broom
Or send a death threat to a boomer
Or DM a girl and groom her
Do a Zoom or find a tumor in your
Here's a healthy breakfast option
You should kill your mom
Here's why women never fuck you
Here's how you can build a bomb
Which Power Ranger are you?
Take this quirky quiz
Obama sent the immigrants to vaccinate your kids

Could I interest you in everything?
All of the time?
A little bit of everything
All of the time
Apathy's a tragedy
And boredom is a crime
Anything and everything
All of the time

Could I interest you in everything?
All of the time?
A little bit of everything
All of the time
Apathy's a tragedy
And boredom is a crime
Anything and everything
All of the time

You know, it wasn't always like this

Not very long ago
Just before your time
Right before the towers fell, circa '99
This was catalogs
Travel blogs
A chat room or two
We set our sights and spent our nights
Waiting
For you, you, insatiable you
Mommy let you use her iPad
You were barely two
And it did all the things
We designed it to do

Now look at you, oh

Look at you, you, you
Unstoppable, watchable
Your time is now
Your inside's out
Honey, how you grew
And if we stick together
Who knows what we'll do
It was always the plan
To put the world in your hand

Hahaha
Could I interest you in everything?
All of the time
A bit of everything
All of the time
Apathy's a tragedy
And boredom is a crime
Anything and everything
All of the time

Could I interest you in everything?
All of the time
A little bit of everything
All of the time
Apathy's a tragedy
And boredom is a crime
Anything and everything
And anything and everything
And anything and everything
And all of the time

---

我操，U 群太牛逼了 . 终于知道 joke3579 那个博里说的东西怎么证了 .

感谢 Athekatelan 和 NaCly_Fish 等人 .

形式 Laplace-Borel 变换

对于 \(F(z)=\sum_{n\ge 0}f_nz^n,\hat F(z)=\sum_{n\ge 0}f_n\frac{z^n}{n!}\)，有：

\[\begin{aligned}F(z)&=\int_0^\infty \hat F(zt)\mathrm e^{-t}\,\mathrm dt\\\hat F(z)&=\frac{1}{2\pi}\int_{-\pi}^\pi F(z\mathrm e^{i\theta })\mathrm e^{\mathrm e^{i\theta}}\,\mathrm d\theta\end{aligned} \]

对于 OGF 和 EGF 相互转换的工具，形式 Laplace-Borel 变换 .

第一个式子：

\[\begin{aligned}F(z)&=\sum_{n\ge0}f_nz^n\\&=\sum_{n\ge0}f_n\dfrac{z^n}{n!}\Gamma(1+n)\\&=\sum_{n\ge0}f_n\dfrac{z^n}{n!}\int_0^\infty t^n\mathrm e^{-t}\mathrm dt\\&=\int_0^\infty\sum_{n\ge0}f_n\dfrac{(zt)^n}{n!}\mathrm e^{-t}\mathrm dt\\&=\int_0^\infty\hat F(zt)\mathrm e^{-t}\mathrm dt\end{aligned} \]

第二个式子：

从右往左推，换元 \(t=\mathrm e^{i\theta}\)：

\[\frac{1}{2\pi}\int_{-\pi}^\pi F(z\mathrm e^{i\theta })\mathrm e^{\mathrm e^{i\theta}}\,\mathrm d\theta=\dfrac1{2\pi\mathrm i}\int_{|t|=1}\dfrac{\mathrm e^t}tF\left(\frac zt\right)\mathrm dt \]

接下来用留数定理即得，过程不难，略去 .

（后一部分是我胡的，如果有问题还请指出）

这其实也告诉我们如果想用形式 Laplace-Borel 变换该怎么用……就和证明差不多积就行了（真的吗？

小五