Chapter 302

"Classmate, would you like to help?" Xu Jie folded his hands together and prayed.

Cheng Nuo rubbed his eyebrows, "Okay, I'll help you take a look, but after reading it, you have to go to bed immediately, if you don't sleep well, even if you review it enough, it's useless." ”

"Of course. Xu Jie handed the book "Preliminary Complex Variable Functions and Operational Differentiation" to Cheng Nuo.

He flipped to a page of the book and pointed to one of the questions, "Ahh That's the question. ”

[Proof: If f∈S, then in Δ:|z|≦1, there is |z|/(1+|z|) ^2≦|f(z)|≤|z|/(1-(x))^2.】

Cheng Nuo glanced at the question briefly and sighed lightly, "Hmm, isn't this Koebe's deviation theorem?"

Xu Jie was confused, "What's that?"

"The Koebe deviation theorem is a defined theorem used to describe the single-leaf function on a unit disk, and the stem of this problem, if I remember correctly, should be the theorem content of Koebe's deviation theorem. In 1944, the mathematician Ke Bei deduced it. Cheng Nuo opened his mouth to explain.

"Why do you know so much?" Xu Jie then asked.

Cheng Nuo shrugged, "I just read a lot of books, and I have read almost a third of the books related to complex variable functions in the school library." ”

A third, that's at least a dozen.

Xu Jie felt a little admiration in his heart, plus some unconvinced, but it didn't show it on his face.

He said, "The point is that you don't know what theorem it is, and the question is about the proof step." ”

Cheng Nuo smiled, "It's simple." He stretched out his hand, "Do you have paper and pen?"

Xu Jie handed Cheng Nuo a pen and paper and looked at Cheng Nuo quietly.

Cheng Nuo didn't rush to write, but asked, "You should know de Branges' theorem, right?"

"This is nature. Xu Jie raised his head proudly, "Although I didn't talk about this theorem in class, I can still say a thing or two based on looking up some extracurricular materials." ”

The de Branges theorem, its predecessor Bieberbach conjecture, was completely solved by the mathematician de Branges in 1912, so it was renamed the de Branges theorem. ”

"Its main content is that if there is a function the power series expansion is f(z)=z+a2z^2+a3z^3+...... , then |an|≦n and the equal sign holds if and only if the function z/(1-z)^2 or its rotation. ”

Bingo!

Cheng Nuo snapped his fingers, "That's true, but what you haven't mentioned is that more than thirty years later, mathematicians deduced the Koebe deviation theorem through the de Branges theorem. ”

"How to derive?" Although Cheng Nuo had already told Xu Jie the theorem needed to derive, he no longer had any clue.

Cheng Nuo raised the pen and slurped on the paper.

First, using de Branges' theorem, the range of f(z) is derived when |z|<1. Since f(0)=0,...... , using the previous one, we get |f(z)|=|∫f(ζ)dζ|≤|z|/(1-z)^2...... Finally, the Koebe deviation theorem is obtained. ”

Xu Jie's expression suddenly realized at this time, his right hand formed a fist and slammed it into the palm of his hand, "Skuyi, so that's the case." Taking de Branges' theorem as the starting point, the general bounded range at |z|<1 is obtained, and finally the reasoning is proved. ”

"Cheng Nuo, I have been taught!" Xu Jie's face was filled with joy, "Although we are not students at the same university, I still hope that you can enter into marriage with me...... Ahh ”

Cheng Nuo stuffed the pen and paper into Xu Jie's hand, "Come on, put away your blessing BUFF, it was safe to win the first prize, but don't be milked to death by you." ”

"Sleep, see you tomorrow. "By the way, remind me again, remember to wake me up!"

Cheng Nuo yawned and fell asleep again.

"I'm not poisoned milk!" Seeing Cheng Nuo, who was already snoring slightly, Xu Jie whispered around his neck.

......

Following. The sky was just dawning.

"Xu Jie, are you crazy, it's only less than six o'clock now!" In room 203 of the hotel, Cheng Nuo's roaring voice echoed.

Once again, Cheng Nuo, who was sleeping soundly, was woken up by Xu Jie.

Next to the bed, Xu Jie had already dressed and packed up, "The plan of the day lies in the morning, classmate Cheng Nuo, the morning is a good time, you can't sleep in vain, come, accompany me to exercise." ”

Cheng Nuo was dragged to get up, he rubbed his sleepy eyes, "Then let's go downstairs for a morning run?"

"No, no. Xu Jie shook his head, "Running is too lo, we should have a unique way of morning exercise." ”

Cheng Nuo: "SO?"

Xu Jie scratched his head and smiled, "So, I created my own set of 'Function Dance'"

The "function dance" invented by Xu Jie is to use the body to construct various basic function curves.

Cheng Nuo was forced to jump together.

y=sinx, a sinusoidal function, bends the arm, and the wave swings.

y=x, linear function, one hand side up, hand side down, 45 degrees.

y=(1-x)^2+3√x^2, cardioid line, specific center!!

............

Sweating profusely and panting.

After breakfast, Cheng Nuo and Xu Jie joined their respective teams separately and took the bus to the final exam venue.

The exam time is the same as the preliminary round, the exam starts at half past nine, and there is three hours of time to answer the questions.

However, in terms of test questions, the final test questions are not only higher in difficulty than the finals, but also add two additional questions.

Each additional question is worth 10 points, and the score is not much, but the difficulty is not small. The two questions were personally tackled by two big names in the field of mathematics.

In previous years, the number of candidates who were able to answer all the last two additional questions correctly on the premise of answering the previous questions could only be described as a handful.

At nine o'clock, more than a dozen people from Tsinghua University came to Modu University, where the test center was located.

At half past nine, the exam officially began.

At ten o'clock, Cheng Nuo completed all the previous questions, leaving only two additional questions.

Additional Question 1: [For the polynomial f(x), denote d(f) to denote its maximum sum and the minimum real root distance of the representation, let n≥2 be the natural number, and find the maximum real number C, so that for any real root ......]

Additional Question 2: [Set X1, X2...... Xn, which are independent and identically distributed random variables, have a common distribution function F(X) and a density function f(x), and now for random variables, X1...... Xn, rearrange ,...... in order of size]

Sure enough, both the entry point of solving the problem and the calculation process are much more troublesome than the previous questions.

But again, for Cheng Nuo, this is not a thing.

The duration of the exam is one hour.

Cheng Nuo walked out of the examination room with a stationery bag.

Professor Lu still sat with Professor Liu, but this time he was smart and did not make any bets with Professor Lu.

"Lao Lu, you are a student, amazing!" Seeing Cheng Nuo coming, Professor Liu said with half emotion and half envy.

Professor Lu laughed and did not answer.

An hour later, Xu Jie also walked out of the examination room in advance.

He came to Professor Liu's side, saw Cheng Nuo sitting on the side, and greeted him with a smile, "Don't say it, this year's questions are really difficult, but fortunately, they are all done." Cheng Nuo, you just came out, right?"

Cheng Nuo spoke, "I'm sorry, I came out an hour ago." ”

Xu Jie: "......"