Summary
Differential Calculus Summary
- Course
- Institution
Summary of differential calculus and Fourier transformations with explanations on how to use the formulas and solve the differential equations.
[Show more]
1 review
By: klaasboterkoek • 3 weeks ago
Seller
Followsterrehoefs
Reviews received
Content preview
Samenvattingen Wiskunde N2, Sint × ) = Odd
cost ✗ ) =
even
even .
Odd - Odd
Fourierreeksen
even .
even = even
Odd Odd =
even
fourierreeks
-
en
even + even
-
-
even
Odd + Odd = Odd
Fourier series : infinite series that re
present period ic function in terms of sines and cosines Fourier Integral : Problems that invdve function that are non periode and are of
interest on the Whale × -
axis
Period ic function : f- IN for AN real ✗ ( except possible at some Points )
de fired > Consider
any period ic function f , (x) of period 2L that can be representeert by
and if there is a positive number p called the period Of f-IN ,
a Fourier series :
for which f- ( ✗ tp ) f- ( x )
=
; or : f- ( ✗ tnp ) =
f-( x ) •
Lo£1 ) s
( cost Wnx ) tbncoslwnx )) with Wn × =
ao +
„=,
an =
Fourier series for Zr period ic function s : f- IN lancoslnxltbnsinlnx)) with :
'
aotn
- = :
'
What happens if L
= , we let -
> as : the series turn into an
Integral
}
r as
ao =
In /
-
r
flxldx >
f- ( x ) =
/[ AIW ) COSIWX ) t Btw ) SINIWX ) dw ] with Fourier
integral coefficients
r
Fourier coefficients 0 as
an , yg„ „ „ „ „ „ „„
given by the AIW ) =
In / f ( v) COSIWV) dv
Euler f- ormulas das
-
„
£ / f- (
"
=L / f- ( sinlnxldx Btw ) v) sinlwv) dv
=
bn x)
- as
-
r
If f- ( x ) is
piecewise continuous in
every finite interval and has a right and
-
left hand der Native at every point and if the a ntegraiexists then f- ( x ) can Derepresented
Trigonometrie System Function zr ( sininx ) , cosinxl )
,
: with a
period of
↳ The
trigonometrie System is
orthogonalon.rs/srlalsoosxszr) ;
by a Fourier
integral .
↳
ftp.ixildx
- as
of any functions in the trig System over that
that is : the Integral two .
At a
poinowhere f- ( x ) is dis continuous , the value of the Fourier integrale qu als the
interval is zero , so that for any integers hand m :
hand limits of
^
"M average of the left -
and right f- ( x ) at that point
{
-
.
°
-
|
r
COSINX ) COSIMX ) DX
= 0 for ( n =/ m) * Kronecker delta QMT
:
Smr 1 AM
Fourier BIW ) -01 integraal of Btw ) dd)
Integral I ff has Fourier
^
cosine : a
Integral representation and is ever, then
{
,
sincnx ) sinlmxld ✗ = 0 for in # m )
then the Fourier integral reduces to a Fourier cosine
as
Integral :
/ AIW { { f- (
i.| interval 0)
f- ( X ) =
) COSIWX ) dw with AIW ) = V) COSIWV ) dv
m)
sinlnx) COSIMX ) DX for in # morn ( Odd Integral symmetrie
= =
= 0 on
.
Fourier integral iff AIW ) even )
sine : has a Fourier
integral representation and is odd , the = 0 / integraal of AIW ) -
_
reducestoa Fourier sine integral :
then the jtourier integral as
Representation by a Fourier series : Jf f- ( x ) is
zr-periodicandpiecewi.se continuous in rsxsr -
f- ( X ) =
| Btw ) sincwx ) dw with BIW ) =
{ { f- In sinlwvldv
and we let f- (x ) have a left and right handed derivative
- -
at each a
Points of that interval , then the Fourier series ( has limit →
approaches real number)
converges a a as
|
coslwx ) kx
f- ( x )
21g e-
Its sum is except at points where f- ( x ) is dis continuous → there the sum of the series
is the
average of the left
-
and
right hand limits of f- ( x ) at ✗o .
Laplace integra / s :
k
'
t wz
dw =
☐
as
f.
wsinlwx )
Fourier series for period 2L &
p
:
dw
7- e-
-
-
=
pz + wz
We set ✓= Î × ,
so dv = Ê dx ,
this
givesus :
Integral trans form
as
f- ( x ) =
aot an cos /Ê × + bnsin 7- × ; with :
: a transformation in the form of an Integral that producers from given function s
function s dep ending on a different variable
a a ,
New .
a. = te / f- klok > can be used in DE 's ,
P DE 's and Integral equations
an
.
_
:
Êfflxicos ( Ex)
L
"
Fourier cosinetransform : concerns even function and is obtained from the Fourier cosine
Integral
bn -
-
E.{ f- als in / 7- ) × "×
Set AIW ) = % Êclw) where c
suggests cosine
Writing v=x
gives
as
:
Even functions : f- c- × ) f- (x ) = ÊCIW ) =
# § FIX ) COSIWX ) dx } Fourier cosinetransform
as
VÉ}
If f- ( x ) is oneven function its Fourier series reduces to a Fourier cosine series :
h f- IN =
ÊCIW ) cos ( wxldx } inverse Fourier cosinetransform :
§ } { f- ÊIW)
as
f- ( X ) =
aotn =,
ancos (% ×
with coefficients ao =L flx ) DX ; an = IN cos
/7- ) DX ×
gives bach fl ✗ I from
Odd function s : f- 1- × ) = -
f- ( x ) Fourier sine trans form : concerns Odd function s and is obtained from the Fourier sine integral
Jf f- ( x ) is an Odd function its Fourier series reducestoa Fourier sine series : set BIW ) =
VÉFÌIW ) where s suggests sine
L
} / f- (7- ) DX Fc / f) Êclw )
as
f- ( x ) =
n= 1
bnsin % × with coefficients bn = IN Sin ×
Writing v. ×
as
gives : Other rotation s : =
0
ÊSIW )
ÊSIW ) VK.f.fi/1siniwxIdx } Fourier sinetransform Fs ( f )
a a =
| glxldx 2/91×1
-
-
Note : = DX for even g
[
L
VÉ Êslw ) sinlwx) DX } inverse Fourier sinetransform
-
0
L
f- IN = :
| hlxldx = 0 for Odd h
gives bach
fl ✗ I from ÊSIW )
* The Fourier transforms
operations : -51 aftbg ) AF / f) Fig )
[
are linear t b
-
=
Half range expansions Spitting the Fourier series
-
: in a Fourier sine series and Fourier cosine series as a
%/ | f- In eiw
" "
-
of the Fourier series :(Best approximation off trigonometrie pdynomial of the same degree N) complex form of the Fourier Integral f/✗ ) dvdw
The Nthpartial by : =
sum a
↳ error is as smalt as
possible cs cs
- -
N
"
Bnsinlnx ))
f-( x )
Aotn (An COSINX ) + Euler for make : e cost ✗ It Isin ( X )
= =
as
= ,
Square of F relative top
ftp.fsf/XIe--imdxJnverseFouriertransforml
Fourier transform ( complex ) ÊIW )
error on -
rsxsr : : =
}
r
| Ik}
"
ÊIW ) ÉN
"
}
' ' '
{
*
/Ao a) ]
'
E- (
f -
F) dx E- [ = r 2
-
t [(An -
an ) + (Bn -
bn)
complex) : f- IN =
dx
-
r n =/
" N
=/ fzdx [Zaoztn Ê f)
'
]
*
[ /anztbn )
'
with E- [
*
E ? E-
*
E:-[
*
BN Other rotation s : 5- If ) f F- (
b n r
If Aóao
only if
=
r > 0 and so and =
;
-
- -
, ,
. . .
.
=, as
r
f IÊIWIÎCIW
-
as
Bessel 's in
equality :
Zaoztn /anztbn )
'
s f / ✗ Îdx Total
energy of a
System
:
,
-
r - as
5- { f- ( x )}
'
For function f Parsevaistheoremholds that
such a is Bessel 's in
equality Fourier transform of the derivative : in 5- { Fix ) }
=
,
as
hdds the equality sign so that it becomes Parseval 's
a
Identity :
=/ / f x-p glpldp
,
n
Control ution f- * g hix ) f- * g) IN flplg / x-p ) dp
te / f- ( ✗ Îdx
•
: = ( =
/ )
Zaoz +
'
lanztbn ) =
- cs -
cs
na r
2nF / f)
5-( f #
g)
=
as
Fcg)
If # g) 1×1=9 ÊIW )
g iw) eindW
ij ij
- as
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller sterrehoefs. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.33. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67474 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now