doey said:Homework Statement
it ask me to integral from 0 to 2pi for 1/[5+4sin(deta)]
Homework Equations
i knoe i nid to find out the pole bu convert sin(deta) into exponential fomat
The Attempt at a Solution
after i solve it until 2z^2+5iz-2 i stucked,can i juz use [-b+-squarot[b^2-4ac]]/2a formula to solve this by juz let my b=5i?
and i get the answer is in negative value,if i do that.i think i might wrong ,any one can help me?
Dick said:Yes. You can use the quadratic equation to solve something like that. But I don't think you've got the right quadratic and I don't know what "i get the answer is in negative value". You really need to show your work.
doey said:∫1/5+4sin(deta) dθ =∫(dz/iz)/[5+4(z/2i-1/2iz)]
= ∫(dz)/[5iz+4(iz^2/2i-iz/2iz)]
= ∫(dz)/[5iz+(4iz^2)/(2i)-4iz/2iz)]
=∫(dz)/[5iz+2z^2-2]
then i nid solve 5iz+2z^2-2 to get my pole,am i wrong? then i will get my root in -0.5i and -2i.
The formula for finding the residue of 1/5+4sin(deta) is Res(1/5+4sin(deta)) = 1/5 - 4/(2πi). This formula is derived from the Cauchy Residue Theorem and is used to calculate the coefficient of the (z-z0)-1 term in the Laurent series of a complex function.
Finding the residue of 1/5+4sin(deta) is important in complex analysis as it allows us to evaluate complex integrals that cannot be solved using traditional methods. The residue theorem states that the value of a complex integral around a closed contour is equal to 2πi times the sum of the residues of the singular points inside the contour. Therefore, finding the residue allows us to evaluate these complex integrals and solve problems in physics, engineering, and other fields.
To find the residue of 1/5+4sin(deta) at a specific point, we use the formula Res(f(z), z=z0) = limz→z0 (z-z0)f(z). This means we take the limit of the function as z approaches the point of interest, multiplied by (z-z0). The resulting value is the residue at that point.
Yes, the residue of 1/5+4sin(deta) can be negative. The residue is a complex number and can have any value, positive or negative. It depends on the function and the point at which the residue is being calculated.
Yes, there is a graphical representation of finding the residue of 1/5+4sin(deta). The Residue Theorem can be visualized using the Cauchy-Goursat Theorem, which states that if a function is analytic in a simply connected region, then any contour integral around that region is equal to zero. This means that the contour integral around a single point is equal to the sum of the contour integrals around the surrounding points, which can be used to find the residues of each point. These residues can then be represented graphically on a contour plot to demonstrate the relationship between the function and its residues.