Lobachevsky's formula: th-cam.com/video/Bq5TB6cZNng/w-d-xo.htmlsi=ZeWTFxTdcQtlN8iy 15 percent off everything using the code MATHS505 on Advanced MathWear: my-store-ef6c0f.creator-spring.com/ Complex analysis lectures: th-cam.com/play/PLVkOfIPb514EP3CjWQQ-JmKpIiNoEUS0k.html&si=rdmwOkQ6Vxg64yrE If you like the videos and would like to support the channel: www.patreon.com/Maths505 You can follow me on Instagram for write ups that come in handy for my videos: instagram.com/maths.505?igshid=MzRlODBiNWFlZA== My LinkedIn: www.linkedin.com/in/kamaal-mirza-86b380252
Beautifully devious, as usual. When you interchange limits it would be good to mention the monotone convergence theorem or the dominated convergence theorem or whatever is needed to justify things.
Maths 505, Thanks for the Integrals. Instead of simply directly ‘solving’ an Integral using any of the ‘Integral-Solving-Techniques’, I would like to first understand how the ‘PLOT’ of the Integrand looks like between the given Limit-Ranges in the problem, and also certain other interesting ranges ; and then, solve the Integral by some form of Geometry-Technique, and then, finally proceed to solve the Integral (using Integration Techniques) : - all this, so that we first get a ‘FEEL’ of WHAT we are actually solving for, atleast in a Fuzzy sense - which might then provide us with ideas of which all Applications, such (/ similar) Integrand Expressions might find use in. Such end-use Applications (ie, the Domains), would then probably automatically ‘lead us’ to a more preferred way of solving the Integral (like for eg, by applying Cauchy’s Residue Theorem, or by Substitutions, or by some other Trick etc). For SIMPLE Examples of this ‘PLOT / Geometric’ method, pl refer to “Mathalysis World - Prakash Pant”’s Vdo : “Solving integrals GEOMETRICALLY in 3 seconds” - for eg, inspect the Screen at 49 s. In this connection, I wonder if you (Maths 505) could re-inspect many of yr Integrals’-Videos, and first, show also the ‘PLOT’ of the Integrand - preferably in various interesting Limit-Ranges {somewhat like the Plots, that Wolfram Alpha (Wα) provides in it’s Output}.
To start with, you (Maths 505) may want to perhaps initially look at these Vdos of yrs first, and then if really practically possible, you might want to edit many of yr other Vdos too ? : {I have had only a quick cursory glance at some of yr vdos ; later, in my FREE time (when ?), I would like to go thru them in DETAIL.} : 1) Yr Vdo : “MOM! I found another golden integral!” - dt Mar 19th 2023 : For eg, inspect the Screen at 3-32 m : Would like to see a Typical PLOT of the Top ‘I’ Integrand and Integral (with Limit Ranges 0 to ∞ ; and any other interesting / revealing Limit-Ranges too) - first, BEFORE applying any Substitution (ie, in the ‘x’ world) ; and later, the PLOT WITH Substitution (ie, in the ‘u’ world). This way, one could quickly, fuzzily, also understand the Mapping correspondence between the diff variables. 2) Yr Vdo : “A RIDICULOUSLY AWESOME INTEGRAL: Ramanujan vs Maths 505” - dt Mar 26th 2023 : For eg, inspect the Screen at 1-19 m : Would like to see a Typical PLOT of the Top ‘I’ Integrand and the Integral, PLUS also some Top 2 Typical / Common Egs of the Infinite Sum expression for f(x) ; PLUS also the relation to the Gamma Fn preferably Geometrically. 3) Yr Vdo : “Feynman's technique is unreasonably OP!” - dt Dec 12th 2023 : For eg, inspect the Screen at 1-58 m : Would like to see a Typical PLOT of the Integral I(α) for typical values of say, α = 1 (as reqd in this Eg), but also at some other interesting values of α (ie, at other than 0 or 1) 4) Yr Vdo : “How Ramanujan proved his master theorem” - dt Dec 16th 2023 : Not just a Typical PLOT of the Top Mellin Xform Integral, but also some 2-3 Typical / Common Egs of the Infinite Sum expression for f(x) - for eg, inspect the Screen at 1-32 m. PLUS also the relation to the Gamma Fn (in this Example), preferably Geometrically.
Some good useful Books : In my younger days, I used to refer the foll 3 books quite often - specially for Maths’ Special Functions, Integral-Expressions and Constants etc : {Providing this List of Books - in the hope that it might be useful for many readers} : {I have also given below the Year of Purchase, because I am not sure if these books might be easily avail now, and from the same Publishers.} : a) “HandBook of Mathematical Functions - With Formulas, Graphs and Mathematical Tables” - Dover Publications, NY - Not sure of the Edition - 9th or 10th ?, but I had purchased in Mar 2004. b) “Table of Integrals, Series, and Products” - 6th Ed - Academic Press - I had purchased in Oct 2004. c) “Mathematical Constants” - by Steven R Finch - RePrint of 2004-05 - Cambridge Univ Press - I had purchased in Aug 2006.
When you said "Wait!", It took me a little bit by surprise, and I thought for a second your next words would, be "make sure to like and subscribe, before you leave."😂
Wait! I thought integral_0->inf(dx*f(x)*sin x/x) = 1/2*integral_0->pi(f(x)*dx), not integral_0->pi/2(f(x)*dx). Therefore the end result should have been I=F(1)=pi/4*(1+ln 2)
@@maths_505 the two formula are equivalent only when f(x) is symmetric wrt pi/2. The difference in end results shows that your version of the formula is no good for this problem.
A cancellation often helps. Like I saw the cosines cancelling out. Even if no cancellation happens, getting rid of the log term was helpful so that's a major factor.
Yes but analysis of the integrand along with it's graph (not to mention Wolfram alpha) point towards convergence. Also, this isn't the first integral I've solved that Mathematica couldn't 😂
@RalphDratman for the integrand: The tangent function has a simple pole for x=(2k+1)π/2 where k is an integer. The log term has a zero at each of those values so the limit of the integrand as x approaches (2k+1)π/2 exists and can be verified to be zero. For x=0, again the limit of tan(x)/x exists and is zero. So there's nothing wrong with the integrand so far. A look at the graph shows that the function and it's oscillations damp out quickly which alleviates fears of divergence. Finally, the result is compared with the output generated by Wolfram alpha. In case Wolfram alpha fails to generate a nice closed form, we can check our closed form numerically. And that sir is how I confirmed that I solved the integral correctly.
Lobachevsky's formula:
th-cam.com/video/Bq5TB6cZNng/w-d-xo.htmlsi=ZeWTFxTdcQtlN8iy
15 percent off everything using the code MATHS505 on Advanced MathWear:
my-store-ef6c0f.creator-spring.com/
Complex analysis lectures:
th-cam.com/play/PLVkOfIPb514EP3CjWQQ-JmKpIiNoEUS0k.html&si=rdmwOkQ6Vxg64yrE
If you like the videos and would like to support the channel:
www.patreon.com/Maths505
You can follow me on Instagram for write ups that come in handy for my videos:
instagram.com/maths.505?igshid=MzRlODBiNWFlZA==
My LinkedIn:
www.linkedin.com/in/kamaal-mirza-86b380252
Hi. Ty for the video
If you don't mind me asking, what's the app you use for this vid? Thanks!
@@YGHF Samsung notes
Amazing. MATLAB's having trouble just integrating this numerically, and you made it look almost easy analytically ... almost.
btw - i like the graphs in the thumbnails, please keep it up.
Beautifully devious, as usual. When you interchange limits it would be good to mention the monotone convergence theorem or the dominated convergence theorem or whatever is needed to justify things.
Or just show the graph in the thumbnail 😂
Hey I just was wondering, what program/materials do you use to draw on your screen like that? the style looks sick
It's a galaxy S6 tab with an s pen
Maths 505, Thanks for the Integrals.
Instead of simply directly ‘solving’ an Integral using any of the ‘Integral-Solving-Techniques’, I would like to first understand how the ‘PLOT’ of the Integrand looks like between the given Limit-Ranges in the problem, and also certain other interesting ranges ;
and then, solve the Integral by some form of Geometry-Technique,
and then, finally proceed to solve the Integral (using Integration Techniques) :
- all this, so that we first get a ‘FEEL’ of WHAT we are actually solving for, atleast in a Fuzzy sense - which might then provide us with ideas of which all Applications, such (/ similar) Integrand Expressions might find use in.
Such end-use Applications (ie, the Domains), would then probably automatically ‘lead us’ to a more preferred way of solving the Integral (like for eg, by applying Cauchy’s Residue Theorem, or by Substitutions, or by some other Trick etc).
For SIMPLE Examples of this ‘PLOT / Geometric’ method, pl refer to “Mathalysis World - Prakash Pant”’s Vdo :
“Solving integrals GEOMETRICALLY in 3 seconds” - for eg, inspect the Screen at 49 s.
In this connection, I wonder if you (Maths 505) could re-inspect many of yr Integrals’-Videos, and first, show also the ‘PLOT’ of the Integrand - preferably in various interesting Limit-Ranges {somewhat like the Plots, that Wolfram Alpha (Wα) provides in it’s Output}.
To start with, you (Maths 505) may want to perhaps initially look at these Vdos of yrs first, and then if really practically possible, you might want to edit many of yr other Vdos too ? :
{I have had only a quick cursory glance at some of yr vdos ; later, in my FREE time (when ?), I would like to go thru them in DETAIL.} :
1) Yr Vdo : “MOM! I found another golden integral!” - dt Mar 19th 2023 :
For eg, inspect the Screen at 3-32 m : Would like to see a Typical PLOT of the Top ‘I’ Integrand and Integral (with Limit Ranges 0 to ∞ ; and any other interesting / revealing Limit-Ranges too) - first, BEFORE applying any Substitution (ie, in the ‘x’ world) ;
and later, the PLOT WITH Substitution (ie, in the ‘u’ world).
This way, one could quickly, fuzzily, also understand the Mapping correspondence between the diff variables.
2) Yr Vdo : “A RIDICULOUSLY AWESOME INTEGRAL: Ramanujan vs Maths 505” - dt Mar 26th 2023 :
For eg, inspect the Screen at 1-19 m : Would like to see a Typical PLOT of the Top ‘I’ Integrand and the Integral, PLUS also some Top 2 Typical / Common Egs of the Infinite Sum expression for f(x) ; PLUS also the relation to the Gamma Fn preferably Geometrically.
3) Yr Vdo : “Feynman's technique is unreasonably OP!” - dt Dec 12th 2023 :
For eg, inspect the Screen at 1-58 m : Would like to see a Typical PLOT of the Integral I(α) for typical values of say, α = 1 (as reqd in this Eg), but also at some other interesting values of α (ie, at other than 0 or 1)
4) Yr Vdo : “How Ramanujan proved his master theorem” - dt Dec 16th 2023 :
Not just a Typical PLOT of the Top Mellin Xform Integral, but also some 2-3 Typical / Common Egs of the Infinite Sum expression for f(x) - for eg, inspect the Screen at 1-32 m.
PLUS also the relation to the Gamma Fn (in this Example), preferably Geometrically.
Some good useful Books : In my younger days, I used to refer the foll 3 books quite often - specially for Maths’ Special Functions, Integral-Expressions and Constants etc :
{Providing this List of Books - in the hope that it might be useful for many readers} :
{I have also given below the Year of Purchase, because I am not sure if these books might be easily avail now, and from the same Publishers.} :
a) “HandBook of Mathematical Functions - With Formulas, Graphs and Mathematical Tables” - Dover Publications, NY - Not sure of the Edition - 9th or 10th ?, but I had purchased in Mar 2004.
b) “Table of Integrals, Series, and Products” - 6th Ed - Academic Press - I had purchased in Oct 2004.
c) “Mathematical Constants” - by Steven R Finch - RePrint of 2004-05 - Cambridge Univ Press - I had purchased in Aug 2006.
very nice approach
It is very interesting solution. Thanks indeed.
When you said "Wait!", It took me a little bit by surprise, and I thought for a second your next words would, be "make sure to like and subscribe, before you leave."😂
Man you scared me 😅
i was about to close the video and i heard someone screem "WAITE"
5:50 missed the opportunity to call it pi-riodic
YOOOOOOOOO!!!!!
Noted for the future!🔥
Non ricordavo quella formula..grazie
I was wondering if you could some day solve some system of DE or some PDE. Anyways you are amazing, and keep the good work mate
Yo that is actually a brilliant idea cuz I haven't done a single video on that ever since the channel started.
How could you put the total derivate under the integral sign, also change it to a partial derivate?
How do you know where to put alpha?
Trial and error
Whay about the integration limits when you do the u = tan(x)?
Excellent!!
You are the best one 😊❤
Thanks bro❤️
Amazing 😊
5:24 bro missed the square term twice , its like when you are so soaked in integration that you forget wth is going on
Wait! I thought integral_0->inf(dx*f(x)*sin x/x) = 1/2*integral_0->pi(f(x)*dx), not integral_0->pi/2(f(x)*dx). Therefore the end result should have been I=F(1)=pi/4*(1+ln 2)
Check out the link in the pinned comment. It's a video proving both versions of the formula (it is probably the best proof video on TH-cam)
@@maths_505 the two formula are equivalent only when f(x) is symmetric wrt pi/2. The difference in end results shows that your version of the formula is no good for this problem.
Absolutely beautiful
The answer is infinity
that value of c really looking like the integration of log(cosx) 🧐
Value of c???
@maths_50 13:49 constant of integration dawg
Isnt it just arcsec at the end? Why arcsech?
Look at a table of integrals and you'll find out why.
It is always surprising (infuriating ) to see everyone use feynman tric without proving its validity while feynman trick can often fails
You mean the conditions needed?
How do you come up with the placement for your alpha in I(alpha)
A cancellation often helps. Like I saw the cosines cancelling out. Even if no cancellation happens, getting rid of the log term was helpful so that's a major factor.
@@maths_505 ye thats fair, thanks
Log2 appears again...
Mathematica complains that
" The integral of tan(x)log(cos(x)+1)/x does not converge on {0,Infinity} "
which is an interesting point, don't you think?
Yes but analysis of the integrand along with it's graph (not to mention Wolfram alpha) point towards convergence. Also, this isn't the first integral I've solved that Mathematica couldn't 😂
@@maths_505 How can you be sure you've got it right? Differentiate?
@@maths_505 And what did you learn from Wolfram Alpha?
@@RalphDratman if I remember correctly, it returned a solution. Though I'm not sure if that was a closed form or not.
@RalphDratman for the integrand:
The tangent function has a simple pole for x=(2k+1)π/2 where k is an integer. The log term has a zero at each of those values so the limit of the integrand as x approaches (2k+1)π/2 exists and can be verified to be zero. For x=0, again the limit of tan(x)/x exists and is zero. So there's nothing wrong with the integrand so far.
A look at the graph shows that the function and it's oscillations damp out quickly which alleviates fears of divergence.
Finally, the result is compared with the output generated by Wolfram alpha. In case Wolfram alpha fails to generate a nice closed form, we can check our closed form numerically.
And that sir is how I confirmed that I solved the integral correctly.
Jajajaja incredibleee... The besttt o f the world
I miss lemniscate ctant
Me too buddy 😥
Thankfully I have one penned down for the holidays🥳
Another day, another integral butchered by Feynman's technique.