A genius? I just love studying.

Chapter 28: Epiphany (Seeking Follow-up Reading)

This is An Chengzhang’s teaching method. He first gives students simple and basic knowledge points, giving them a taste of the sweetness so that they will become interested in the subject.

Then we raise the level of knowledge through questions, allowing students to explore the application skills of these knowledge points by themselves, and finally integrate and truly learn them!

After explaining two more questions, the second class ended.

In the third class, An Chengzhang did not explain the questions, but distributed some materials for the students to study on their own.

Ding-ling-ling, the get out of class bell rings.
The students stood up with sighs, packed their bags tiredly and prepared to go home.

They now feel that analytic geometry is not that simple.

Chen Hui did not move, and continued to concentrate on studying the contents of the textbook.

The third class in the evening ends at 9:20, and he usually studies until 10 o'clock before going home.

Although his insight has improved to a level beyond that of a high school student, learning mathematical analysis is not easy.

An Chengzhang didn't leave either. He just sat on the podium and read a book.

The classmates left one after another, and the classroom became quiet, with only the sound of two people flipping through books.

sizzle...

Suddenly, a rapid ringing sounded.

Chen Hui came back to his senses from studying and looked back at the big clock hanging on the back wall of the classroom. It was already nine fifty-five and it was time for him to leave.

When he stood up to pack his schoolbag, he noticed An Chengzhang reading a book on the podium.

An Chengzhang also looked up and smiled at him, then stood up and walked towards Chen Hui. "Are you teaching yourself mathematical analysis?"

"Yeah." Chen Hui nodded.

"What did you learn?"

"Euler Integrals and Fourier Series Expansions."

"Oh? Euler integral and Fourier series expansion?"

An Chengzhang raised his eyebrows. Euler's formula can be used in Fourier series expansion, but Euler integral has nothing to do with Fourier series.

He now had some doubts as to whether this boy hadn't learned it at all.

"Um,"

Chen Hui's brows were also furrowed. "I didn't understand some parts of Euler's integral, so I skipped to the back and learned the Fourier series expansion."

An Chengzhang suddenly understood, and his brows relaxed. "What don't you understand?"

Since the teacher asked the question, Chen Hui was not in a hurry to leave. He took out the textbook from his schoolbag and opened it to the chapter on Euler integrals.

"Here, the textbook says that through the gamma function and the beta function, we can obviously get B(p,q)=Γ(p)Γ(q)/Γ(p+q)."

Chen Hui pointed at the textbook, his tone somewhat resentful, "This relationship is not so obvious..."

An Chengzhang smiled. It was obviously easy to get. By the same token... their classmates in the mathematics department had been tortured by this kind of thing a lot.

Recalling the youthful years, An Chengzhang couldn't help but smile.

"Write it down for me if you don't understand."

Putting away his smile, An Chengzhang asked again.

Logically speaking, as the coach of the digital competition team, he should stop Chen Hui from doing this unprofessional behavior.

It is right to learn more advanced mathematics knowledge, but it will help very little in competitions. University mathematics knowledge cannot be used in high school competitions. It is better to use this time to study competition skills. Maybe you can get a few more points. Every point you get in the competition is of great significance.

But he is not an orthodox competition coach, otherwise he would not use the current teaching method, but would directly take out the Olympic blue book and read from it.

He hopes to stimulate students' interest in mathematics through competition training rather than simply for the sake of getting high scores.

Chen Hui sat down naturally, took out his draft book, and began to write and draw.

First, write the expressions of Γ(p) and Γ(q) with respect to s and t, then multiply the two expressions and set a set Ω={(s, t)|s>0, t>0}, so that the multiplication of the two integrals can be turned into a double integral on Ω.

Seeing this, An Chengzhang nodded. The fact that Chen Hui could write this far showed that he had a solid grasp of the calculus content. It seemed that he had really learned this calculus book. Chen Hui stopped writing here and turned to write out the expression for Γ(p+q), then looked up at Teacher An.

He wrote this process very quickly, and it was obvious that he had performed many calculations and attempts before.

"The expression Γ(p)Γ(q) can be further processed. How do we usually solve double integrals?"

An Chengzhang did not give the answer directly, but began to guide.

"Use repeated integration, but I tried, and I still can't get the final relationship."

Chen Hui frowned.

"Here we need to use a trick, using polar coordinates to convert variables, let s = rcosθ, t = rsinθ..."

An Chengzhang stopped there.

Chen Hui had already picked up his pen again. Although he didn't see any use in doing so, he might as well give it a try.

After substitution, we can calculate the range of r and θ through the range of s and t, and then convert the double integral into a cumulative integral.

The moment he wrote the expression, Chen Hui suddenly understood!
Finally, we get the result of multiplying the two parts, and the first expression is exactly the beta function B(p, q), and the second expression is Γ(p+q)!

Is this proven?

It's that simple?
Just like water flowing through a blocked ditch, after a brief moment of shock, Chen Hui felt relaxed all over.

[Your math level increased from level 2 1% to 2%]

At the same time, a barrage of words flashed before his eyes.

My math level has actually improved again!
Chen Hui's eyes fixed.

His math level just broke through yesterday, and that was the result of a week of intensive study. But now, he only uses his spare time to study math, and in less than two days, his proficiency has increased by another 1%!

Without much doubt, Chen Hui understood the reason.

This is the importance of teachers!
If a geography teacher can double the speed of proficiency improvement, then a math teacher can certainly do the same!

He didn't have this feeling when he was listening to the class just now, because what Teacher An said was of no help to him, so his proficiency naturally didn't change significantly. But now, what Teacher An said was exactly what he needed, so his math proficiency naturally skyrocketed.

Of course, this is also because he himself had thought deeply about this problem and had accumulated more than a week of experience, so his proficiency directly increased by 1% after solving this problem.

This feeling is somewhat similar to what Li Hai said about a cultivator breaking through a bottleneck.

An Chengzhang also nodded secretly. No teacher would dislike a student who understood things so quickly.

After thinking for a while, he picked up Chen Hui's pen, lay on the desk, and began to write and draw on Chen Hui's draft paper.
【给定欧拉积分中贝塔函数B(p，q)=∫x^(p1)(1x)^(q1)dx，其中 p>0且 q>0。

(1) Calculate...

"I'll leave you two questions. Do them first. If you don't understand, ask me."

After quickly writing down two questions, An Chengzhang put away his pen and stood up. He raised his wrist to check the time and said, "Let's go. We'll stop here for today. You can look at it tomorrow."

An Chengzhang pointed to his head and said, "You need to rest early. For math students, the brain is very important. Only when you have a good rest can you have good inspiration!"

"Come on, I'll drive you home!"

The school gate was already closed at this time, but An Chengzhang was a teacher, so he still had some privileges.

Chen Hui was silent. He didn't dare to say that he had to go back and learn three more points.

He knew that what Teacher An said made sense, but he felt that even if he studied until three in the morning, he was still in good spirits, which was different from other students.

But there is more to his difference from other students than just this. He did not delve into it deeply, which is a good thing. Since he can learn, he should learn more. Knowledge is power!
"Mathematics is still very interesting. Gauss, Euler, Riemann..."

While walking on the road, An Chengzhang chatted with Chen Hui. As a graduate student in the Department of Mathematics, he still had a sacred belief in mathematics.

"Don't just focus on moving forward and learning. Sometimes you can also learn about these talented mathematicians. Perhaps their lives can inspire you."

Of course, An Chengzhang was the one doing most of the talking, while Chen Hui listened silently. "For example, when I learned about Euler's integral, I looked into Euler's life, his achievements in mathematics, and the formulas and theorems he proposed."

An Chengzhang became more and more excited as he spoke, his face full of radiance. "These things will be buried in your heart like seeds, and when you need them, they will suddenly take root, sprout, and bear fruit. This feeling is really wonderful!"

(End of this chapter)