I'm posting some revised versions of these older videos. The revisions are cleaned up and the graphics should be easer to read. You may want to view the revised versions instead of the originals.
@Oxydox Interesting Comment! Checking some handy references gives some integral formulas involving Jn, In,Kn,Yn and Gammas. Abramowitz and Stegun give Jn(x) as a contour integral of a ratio of two Gamma's and another factor. Going the other way, Gradshtyn & Ryzhik give the product of two Gamma's as the integral of three J's divided by the product of two Associate Legendre Functions. Integral formulas for example: A&S p. 486 II.4.12-22 G&R p.672-673.
@palui Ah yes! Interesting question. How is it possible that the integral has the same value for n=0 AND n=1? Do you have any suggestions how we might answer this question? I'm thinking of plotting the integrands over the range of integration. That would be one place to start. More...later. Good thinking.
@Mathview I see. I confused myself. I was trying to take the area under the curve of the integral (the Gamma function). Instead, I should be taking the area under the curve of the integrand. The Gamma function is connecting a smooth curve through the factorial points, and I was looking at the area under THAT curve. I should have been looking at the areas under the integrand for n=0,1,2...
@Monkeyshex There actually is a good reason for 0!=1. You get it because the Gamma function is continuous. There is a similar situation with raising 10 to the 0 power. Just look at the function 10^x and take the limit as x goes to 0. 10^x is a continuous function of x and 10^0=1. A little more tricky is f(x,y)= x^y. i.e. x to the y power. What do you get for x =0 and y=0? Can you get the answer by taking the limits? Does it depend on which limit you take first?
Ooops...aka erratum/ Unlike most "special fuctions" encountered in applied mathematics, the Gamma Function has the unusual property that it is NOT the solution of a differential equation. This erratum does not change any of the results in this video.
This would be much more legible if it was typed using Scientific Notebook or Workplace, and made using screen capture software like Camtasia Studio or Screencorder. (I have no connection with any of those companies)
It seems that the black background provides more contrast and makes for better viewing. I also increased the pixel resolution of the graphic board in the recent ones. I hope that helps.
gahh, my eyes!
swartschkalle 3 months ago 3
@swartschkalle Sorry!
Mathview 3 months ago
use some other colors FUCK
omaregb 7 months ago
I'm posting some revised versions of these older videos. The revisions are cleaned up and the graphics should be easer to read. You may want to view the revised versions instead of the originals.
Mathview 1 year ago
WTH
XDROBLOXGUYXD 1 year ago
I kinda wish you would use white background with black text. the pink on purple is a little obscure...
AgentMidnight 1 year ago
@AgentMidnight I agree with you. More recent videos use a deeper black background and contrasting hues for the pen tool.
Mathview 1 year ago
@Oxydox Interesting Comment! Checking some handy references gives some integral formulas involving Jn, In,Kn,Yn and Gammas. Abramowitz and Stegun give Jn(x) as a contour integral of a ratio of two Gamma's and another factor. Going the other way, Gradshtyn & Ryzhik give the product of two Gamma's as the integral of three J's divided by the product of two Associate Legendre Functions. Integral formulas for example: A&S p. 486 II.4.12-22 G&R p.672-673.
Mathview 1 year ago
0! =1 and 1! = 1, so how is it that the area under the curve at both n=0 and n=1 are the same?
palui 1 year ago
@palui Ah yes! Interesting question. How is it possible that the integral has the same value for n=0 AND n=1? Do you have any suggestions how we might answer this question? I'm thinking of plotting the integrands over the range of integration. That would be one place to start. More...later. Good thinking.
Mathview 1 year ago
@Mathview I see. I confused myself. I was trying to take the area under the curve of the integral (the Gamma function). Instead, I should be taking the area under the curve of the integrand. The Gamma function is connecting a smooth curve through the factorial points, and I was looking at the area under THAT curve. I should have been looking at the areas under the integrand for n=0,1,2...
palui 1 year ago
@Mathview 0! doesnt even make sense, it's just defined to be 1 for convenience. I wouldn't worry about it.
Monkeyshex 1 year ago
@Monkeyshex There actually is a good reason for 0!=1. You get it because the Gamma function is continuous. There is a similar situation with raising 10 to the 0 power. Just look at the function 10^x and take the limit as x goes to 0. 10^x is a continuous function of x and 10^0=1. A little more tricky is f(x,y)= x^y. i.e. x to the y power. What do you get for x =0 and y=0? Can you get the answer by taking the limits? Does it depend on which limit you take first?
Mathview 1 year ago
Ooops...aka erratum/ Unlike most "special fuctions" encountered in applied mathematics, the Gamma Function has the unusual property that it is NOT the solution of a differential equation. This erratum does not change any of the results in this video.
Mathview 1 year ago
Nice video,
What kind of mouse are you using to write those math smbols?
dragonbrave48 1 year ago
This would be much more legible if it was typed using Scientific Notebook or Workplace, and made using screen capture software like Camtasia Studio or Screencorder. (I have no connection with any of those companies)
NatalieKehr 2 years ago
never use that horrible pink colour again.
rohypnol55 2 years ago 14
thx for the video
kompabt 2 years ago
It seems that the black background provides more contrast and makes for better viewing. I also increased the pixel resolution of the graphic board in the recent ones. I hope that helps.
Mathview 3 years ago