- #1
Jamin2112
- 986
- 12
Homework Statement
As in thread title.
Homework Equations
Residue Theorem.
The Attempt at a Solution
I just need help figuring out the circle C I'll be using. Suggestions?
vela said:What does the presence of z2/3 tell you?
vela said:Yes, other than that. In particular, what's the effect of the fractional power?
vela said:Right. Do you know what a branch point and a branch cut are?
vela said:It's more that you want to avoid crossing the branch cut than avoiding z=0, and you obviously want a piece or pieces of the contour to correspond to the original integral.
vela said:No, that's too complicated. Take a look at http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28IV.29_.E2.80.93_branch_cuts.
Also, rewrite the integrand as
$$\frac{z^{1/3}}{z(z+1)}$$to make it clear how to calculate the residue at z=0.
vela said:Doesn't the answer to that question depend on which way Pacman is moving?
vela said:Start by taking the keyhole contour and break it into four pieces and evaluate the line integral for each piece.
The integral ∫dx/((x^(2/3)(x+1)) over [0,∞] represents the area under the curve of the function 1/((x^(2/3)(x+1)) on the interval [0,∞]. It is a measure of the total amount of space between the curve and the x-axis.
The process for evaluating this integral involves using integration techniques such as substitution, integration by parts, or partial fractions. The specific technique used depends on the complexity of the function 1/((x^(2/3)(x+1)) and the comfort level of the scientist with various integration methods.
The interval [0,∞] indicates the range of values over which the function 1/((x^(2/3)(x+1)) is being integrated. In this case, the integral is being evaluated from 0 to infinity, meaning that the area under the curve is being measured for all values of x greater than or equal to 0.
Yes, this integral can be solved analytically using integration techniques. However, it may require multiple steps and may not have a closed form solution.
This integral can be used in various fields of science, such as physics, engineering, and economics, to calculate the total amount of a quantity or to determine the average value of a function. It can also be used to solve differential equations and to model real-world phenomena.