Very interesting integral ! Also , the appearance of the Bessel and Struve functions show how this integral is related to higher transcendental functions , which play an important role e.g. in physics.
Love the algebraic manipulation but the solution does feel unsatisfying. You've replaced an integral with first the gamma function then by a pair of exotic functions which are themselves defined as integrals or infinite series. Perfectly acceptable for an advanced algebra or calculus class, but I don't see how it advances our understanding or interpretation of the answer. Maybe a bit longer exposition at the end would have helped. Perhaps a graph or a description of these functions' properties to explain to us how to interpret the answer. Hope this is helpful. In the end nothing wrong with your math, but the video doesn't leave me with any knowledge once it's over.
I agree This is my first video involving exotic stuff so it was kinda experimental. I'm gonna make longer videos like these and separate videos playing with special functions on my whiteboard. Thanks for your wonderful insight as always
@@maths_505 much appreciated! I think explaining the properties of those functions would allow us to interpret the answer in a novel way vs how we interpret the original problem. Then we can develop an insight into how those functions work. I think back to how complex analysis first introduced Euler's formula to give you a new interpretation of the sin and cos functions. Something similar for these exotic functions would provide that same level of new insight. Looking forward to more of your videos. All the best!
@@zunaidparker thank you so much and yes I agree with your perspective. Many problems in physics are solved using such insights and so I'm inclined to to venture into that area of RUclips mathematics
for a moment there I thought we're going to add by e^isin(x) and find a relation between e^cos(x) and the real part of the exponential...exponential function... which isn't easier to solve.... idk I think I was thinking about e^(cos(x))cos(cos(x)) (from bprp). but I'll leave that as an exercise for santa, homie be living on the north polar coordinates.
Interesting result! But I have something to point out, which is the fact that at 4:06 you used the recursive Gamma property ending up with k in the denominator, which is kind of problematic since we're summing from k=0 to infinity. How does the result still hold even if we take account of this fact?
I'm gonna do more videos on special functions and higher transcendentals where I'm gonna discuss their properties using graphs etc. Discussing the approximate values of convergence here would've made the video too long so I'll discuss this separately. Till then you can Google numerical approximations for the answer.
Very interesting integral ! Also , the appearance of the Bessel and Struve functions show how this integral is related to higher transcendental functions , which play an important role e.g. in physics.
Indeed
I'm gonna do alot more videos on special functions and their properties too
Love the algebraic manipulation but the solution does feel unsatisfying. You've replaced an integral with first the gamma function then by a pair of exotic functions which are themselves defined as integrals or infinite series. Perfectly acceptable for an advanced algebra or calculus class, but I don't see how it advances our understanding or interpretation of the answer.
Maybe a bit longer exposition at the end would have helped. Perhaps a graph or a description of these functions' properties to explain to us how to interpret the answer.
Hope this is helpful. In the end nothing wrong with your math, but the video doesn't leave me with any knowledge once it's over.
I agree
This is my first video involving exotic stuff so it was kinda experimental. I'm gonna make longer videos like these and separate videos playing with special functions on my whiteboard. Thanks for your wonderful insight as always
@@maths_505 much appreciated! I think explaining the properties of those functions would allow us to interpret the answer in a novel way vs how we interpret the original problem. Then we can develop an insight into how those functions work.
I think back to how complex analysis first introduced Euler's formula to give you a new interpretation of the sin and cos functions. Something similar for these exotic functions would provide that same level of new insight.
Looking forward to more of your videos. All the best!
@@zunaidparker thank you so much and yes I agree with your perspective.
Many problems in physics are solved using such insights and so I'm inclined to to venture into that area of RUclips mathematics
@@maths_505 even better if they have a physics application. Then you can relate it back to the relevant motion, force etc. Looking forward to it!
This is a non-elementary function I didn’t know😢😢. But for compensate this I used riemann’s sum
for a moment there I thought we're going to add by e^isin(x) and find a relation between e^cos(x) and the real part of the exponential...exponential function... which isn't easier to solve.... idk I think I was thinking about e^(cos(x))cos(cos(x)) (from bprp). but I'll leave that as an exercise for santa, homie be living on the north polar coordinates.
That's actually quite an easy integral that can be solved using Feynman's technique....you know what I'll do that in my next video
@@maths_505 aww man what will be left for santa? once he converges to the e-neighborhood?
I think Santa might have a few infinite series in mind this time around
I have to remember to drink my coffee before I watch your videos first thing in the morning. WOW!!! What a slap in the face. Great video!!!!
My friend the fact that you watched this video first thing in the morning is heartwarming.
Happy holidays !
I've a question: Which Editor (or App) did you use to create this video? Thank you.
He made a community post about this iirc
Interesting result! But I have something to point out, which is the fact that at 4:06 you used the recursive Gamma property ending up with k in the denominator, which is kind of problematic since we're summing from k=0 to infinity. How does the result still hold even if we take account of this fact?
I know my handwriting is horrible 🤣🤣
When I saw the reduction formula, I instantly thought about the video « how to teach calculus with sarcasm » haha
Hello integration by parts haters
Reduction for cosine can be derived by parts
Int(cos^n(x),x)=Int(cos(x)cos^(n-1)(x),x)
Int(cos(x)cos^(n-1)(x),x) = sin(x)cos^(n-1)(x)-Int(sin(x)(n-1)cos^(n-2)(x)(-sin(x)),x)
Int(cos^n(x),x) = sin(x)cos^(n-1)(x) + (n-1)Int(cos^(n-2)(x)sin^2(x),x)
Int(cos^n(x),x) = sin(x)cos^(n-1)(x) + (n-1)Int(cos^(n-2)(x)(1-cos^2(x)),x)
Int(cos^n(x),x) = sin(x)cos^(n-1)(x) + (n-1)Int(cos^(n-2)(x),x) - (n-1)Int(cos^n(x),x)
(1+ (n-1))Int(cos^n(x),x) = sin(x)cos^(n-1)(x) + (n-1)Int(cos^(n-2)(x),x)
n Int(cos^n(x),x) = sin(x)cos^(n-1)(x) + (n-1)Int(cos^(n-2)(x),x)
Int(cos^n(x),x) = 1/n sin(x)cos^(n-1)(x) + (n-1)/n Int(cos^(n-2)(x),x)
This reduction formula will make video understandable for those who do not know special functions like Gamma function
You can use thé Wallis' integrals to get a more "explicit" result
I just made notes for exactly that😂
what does it converge to?
I'm gonna do more videos on special functions and higher transcendentals where I'm gonna discuss their properties using graphs etc.
Discussing the approximate values of convergence here would've made the video too long so I'll discuss this separately. Till then you can Google numerical approximations for the answer.