MMG700 tentauppgift .pdf

File information


Original filename: MMG700 tentauppgift.pdf

This PDF 1.5 document has been generated by TeX / pdfTeX-1.40.15, and has been sent on pdf-archive.com on 05/07/2016 at 18:58, from IP address 79.136.x.x. The current document download page has been viewed 293 times.
File size: 107 KB (1 page).
Privacy: public file


Download original PDF file


MMG700 tentauppgift.pdf (PDF, 107 KB)


Share on social networks



Link to this file download page



Document preview


Tentauppgift, komplexanalys

Calculate the integral



Z
0

cos(2x)
dx
x2 + 2

Solution: First note that the integrand is even, hence
Z



0

Integrate f (z) =

e2iz
z 2 +2

1
cos(2x)
dx =
x2 + 2
2



−∞

cos(2x)
1
dx =
x2 + 2
2

Z



−∞


< e2ix
dx
x2 + 2

along the curve ΓR

The function f has the single pole

res{f,

Z





2i inside ΓR (given R >



2), and


√ 
e−2 2
e2iz 
e2iz


2i} = lim
z

2i
=
lim
=


2
2i
2 2i
z→ 2i z + 2
z→ 2i z +

hence



πe−2
f dz = √
2
ΓR

Z

2




By Jordan, the integral along ΓR \[−R, R] that is, the integral along Reiθ θ ∈ (0, π) equals
0, and since e2iz = cos(2z) + i sin(2z) where sin(2z) is odd, we see that


Z
[−R,R]

cos(2z)
πe−2

dz
=
z2 + 2
2

2

Z
−i
[−R,R]

|
That is,
Z
0



=0



πe−2 2
cos(2x)

dx
=
x2 + 2
2 2

1

sin(2z)
dz
z2 + 2
{z
}


Document preview MMG700 tentauppgift.pdf - page 1/1


Related documents


mmg700 tentauppgift
calculus at it s finest
diff eq
math words
a4 w18 w picture
uniqueness in lie theory

Link to this page


Permanent link

Use the permanent link to the download page to share your document on Facebook, Twitter, LinkedIn, or directly with a contact by e-Mail, Messenger, Whatsapp, Line..

Short link

Use the short link to share your document on Twitter or by text message (SMS)

HTML Code

Copy the following HTML code to share your document on a Website or Blog

QR Code

QR Code link to PDF file MMG700 tentauppgift.pdf