10:30 Reminds me of an old math joke I learned in college A mathematician and an engineer are working on math homework together involving 7-dimensional mathematics. The engineer is struggling to really understand the subject matter, but the mathematician is zipping through it The engineer says "how is this so easy for you? I mean, 3 dimensions is easy, but 7? I just can't wrap my head around it" The mathematician replies: oh it's easy. I just think of it like an N-dimensional problem, and set N to 7"
There are math channels on youtube where this video would be three times longer. I hate long and unnecessarily detailed explanations of supposedly obvious techniques. This video is a nice exception, the right amount of (giant) steps a mathematician needs to take to solve a problem, any problem. The emphasis should be on the ideas, not the techniques to implement them. I followed everything with ease, although I can't figure it out off the top of my head. Now I know. Thank you, the recreational mathematics is the best.
I thought that, too. I threw the integrand into Wolfram Alpha to get a Taylor series at x=0. Then you can integrate term by term. Problem is...then you get an infinite series as an answer, which you can't just look at and say....hmmm that looks like ln(2)*PI/8.
The way I solved it (method 4): I expanded ln(1+x) into its Taylor series. Took way longer and had to use a lot more tricks. The integrals themselves became quick and easy (albeit infinitely many), but then there was the summation part. Had to use some weird manipulations, and ended up using these two specific sums over and over again 1/2 - 1/4 + 1/6 - 1/8 + ... = ln(2)/2 1 - 1/3 + 1/5 - 1/7 + ... = pi/4 What matters is at the end I got to pi * ln(2) / 8. So it's all good
@6:04 doesn’t continuity only guarantee this in the case of a bounded interval, but in general we would need the integrals to be integrable to do the swap?
great video! there's also another method involving some rather tedious work but also works for Serret's Integral 1. substitute x = tan(theta) 2. cancel out sec^2 3. combine the inner term of the natural log 4. utilize the harmonic addition for its numerator 5. utilize log properties twice to get three terms (the first term directly gives the final answer) 6. cancel out the remaining cosine terms using King's integral
I may sound a little arrogant but these intergrals are taught in high schools in India. This particular integral was solved by the first method you used .
Dr trefor Thank you for your helpful videos! I have an exam on Thursday and was wondering if you could revisit or explain more about the equivalence of propositions in another video. Your teaching style really helps me understand better. Thank you so much!"
@DrTrefor Thank you for your reply! I’ve watched the video in your playlist, but I’m still confused about how to determine the truth value of equivalence propositions step by step. Could you explain this part in more detail or provide an example? It would really help clear up my confusion.
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10:30
Reminds me of an old math joke I learned in college
A mathematician and an engineer are working on math homework together involving 7-dimensional mathematics. The engineer is struggling to really understand the subject matter, but the mathematician is zipping through it
The engineer says "how is this so easy for you? I mean, 3 dimensions is easy, but 7? I just can't wrap my head around it"
The mathematician replies: oh it's easy. I just think of it like an N-dimensional problem, and set N to 7"
I like how you explain why you choose each step. The less mystery the better education!
from Morocco thank you for this clear complete full proofs
You can use Residue theory
There are math channels on youtube where this video would be three times longer. I hate long and unnecessarily detailed explanations of supposedly obvious techniques. This video is a nice exception, the right amount of (giant) steps a mathematician needs to take to solve a problem, any problem. The emphasis should be on the ideas, not the techniques to implement them.
I followed everything with ease, although I can't figure it out off the top of my head. Now I know.
Thank you, the recreational mathematics is the best.
Very nice explanation professor! Thank you!
What about complex analysis?
KINDA HARD SIR.
can absolutely be done this way!
@DrTrefor Do you have any videos on complex analysis?
So impressive. Use of basic identities and well known rules. Adding the integrals and logs.
it's a cool one!
The denominator screams a geometric series with -x^2 as the common ratio, which absolute value is less than 1.
I thought that, too. I threw the integrand into Wolfram Alpha to get a Taylor series at x=0. Then you can integrate term by term. Problem is...then you get an infinite series as an answer, which you can't just look at and say....hmmm that looks like ln(2)*PI/8.
The first method was the simplest for me to understand.
ya I think that is the most "elementary"
The way I solved it (method 4): I expanded ln(1+x) into its Taylor series.
Took way longer and had to use a lot more tricks. The integrals themselves became quick and easy (albeit infinitely many), but then there was the summation part.
Had to use some weird manipulations, and ended up using these two specific sums over and over again
1/2 - 1/4 + 1/6 - 1/8 + ... = ln(2)/2
1 - 1/3 + 1/5 - 1/7 + ... = pi/4
What matters is at the end I got to pi * ln(2) / 8. So it's all good
@6:04 doesn’t continuity only guarantee this in the case of a bounded interval, but in general we would need the integrals to be integrable to do the swap?
ya for sure, our region is the square [0,1]\times[0,1] so it works here
Do you have a video on your set-up?
What softwares and set up are you using for this video?
great video! there's also another method involving some rather tedious work but also works for Serret's Integral
1. substitute x = tan(theta)
2. cancel out sec^2
3. combine the inner term of the natural log
4. utilize the harmonic addition for its numerator
5. utilize log properties twice to get three terms (the first term directly gives the final answer)
6. cancel out the remaining cosine terms using King's integral
i just noticed this method was mentioned by @Unoqualunque in the stack exchange post you linked, oh well!
oh very nice one!
By 3., you probably mean that one writes 1 + tan(theta) = (cos(theta) + sin(theta))/cos(theta)?
But what does "harmonic addition" in 4. mean?
@@bjornfeuerbacher5514also wondering
@@bjornfeuerbacher5514 sorry i wrote this at 2 am, i meant the harmonic addition theorem
It can also be solved by placing x=(1-t)/(1+t)
Yup, that's a great method:)
Very smart mobius transform
I may sound a little arrogant but these intergrals are taught in high schools in India. This particular integral was solved by the first method you used .
I opened this and this video started with german synced voice, damn that sounded super shit
youtube is tripping
Math is the explain of the grey hair and young face
Dr trefor Thank you for your helpful videos! I have an exam on Thursday and was wondering if you could revisit or explain more about the equivalence of propositions in another video. Your teaching style really helps me understand better. Thank you so much!"
I doubt I can do that by thursday! But that is a good idea:) You've seen my discrete math playlist? I have a few videos there that might help.
@DrTrefor Thank you for your reply! I’ve watched the video in your playlist, but I’m still confused about how to determine the truth value of equivalence propositions step by step. Could you explain this part in more detail or provide an example? It would really help clear up my confusion.
Feyman's trick is also what I did
not satisfying, just mountains more confusion. ...- a struggling math student.