MHB Change of variables/ Transformations part 2

Joe20
Messages
53
Reaction score
1
I am not sure how I should set my u and v expressions into the u-v plane for this question.
How should I look at the expression to set u and v expressions?
 

Attachments

  • ca2.png
    ca2.png
    8 KB · Views: 113
Physics news on Phys.org
Alexis87 said:
I am not sure how I should set my u and v expressions into the u-v plane for this question.
How should I look at the expression to set u and v expressions?
You could start by letting $x = 2u$ and $y = 3v$. You might then want to make a further change, from cartesian to polar coordinates.
 
You probably know that parametric equations for a circle with radius r, centered at (0, 0), x^2+ y^2= r^2, are x= r cos(t), y= r sin(t) because x^2+ y^2= r^2cos^2(t)+ r^2 sin^2(t)= r^2(cos^2(t)+ sin^2(t))= r^2.

It should not be too much of a "jump" to see that parametric equations for the ellipse, with axes of length a and b in the x and y direction, respectively, x^2/a^2+ y^2/b^2= 1, are x= a cos(t), y= b sin(t).
 
For original Zeta function, ζ(s)=1+1/2^s+1/3^s+1/4^s+... =1+e^(-slog2)+e^(-slog3)+e^(-slog4)+... , Re(s)>1 Riemann extended the Zeta function to the region where s≠1 using analytical extension. New Zeta function is in the form of contour integration, which appears simple but is actually more inconvenient to analyze than the original Zeta function. The original Zeta function already contains all the information about the distribution of prime numbers. So we only handle with original Zeta...

Similar threads

Replies
3
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
6
Views
1K
Replies
1
Views
2K
Replies
29
Views
4K
Replies
2
Views
1K
Back
Top