Math Methods - astronomy.nmsu.edu
Transcript of Math Methods - astronomy.nmsu.edu
Accuracy of series approximations :
Remaindervan flat ILENE,
,
Rnlnh.flnl-E.fm/0l.xIf.Mms of the series of order higher then n
.
Conmy if fino.
I Rnln ) ) =o
flu )= Tulu ) +12 t '
the
"myaµnE€iHmt#µProof: By induction
.
For
÷2^1 ntffh - T.la ) = Hnl . flat - flk ) ( × - al
Gnsidh
f×#Hf" Hldta
Use integration by parts
fudv = no - frdu with ni × - t;
du = - dt
drigittldt ; v.-
f' HIThaiTxttf"Hdt=k .ttf 'H ftp.fxfttht
= #) . Ix. a) flla) + fk ) - ft )
An '- fftdtlx. a
)glqD=R
,he ) =fY*Hf" Halt
÷n )so
. the theorem is proved for n 't.
Suppose how that it is tvl for n=k.
Rnln ) it ,fax lx. Hk f
" 4 "it ) dt
lends show that if this true for nik,
than it is he forn .
-
Ktat '×tia÷yf×#tY" f
' 'm 'Hat
Again use integration by parts
fudr = no - rfduwith
u= it . t )" '
dr =/" +2 '
A )
du : - lkthktlkdt r= fkt"
A)£attn , [ EH" 'f"
" '
Hat =
= with 't '"
't '"" ''ABE '
that, Eh.tit
' " ' '
that
= to -
Fi :' '' ' at "f' " "
late ,
ftp.tihf" " "
Hdt
1kt
II li ) ( n . al" "
i Rklx )
Ht1) !
=P In ) - f" " " la ) ( n . al
" "
K
←1 ) !
= Hm . t,
H - fill In . a" "
Ht1) !
=tm . fix'tftp.IHK.at" "
)= fth - Tw ,
(a) = 12114+11×1
Proved that it is we for net and that.
if truefor n= K
,it will be true for n = 1<+1
.So
, the theorem is
Proved by induction .
A more useful form of the theorem : =iiRH=fK¥t" ft"Yt ) dt
'
Makingthe interval as small as possible
The derivative approaches a constant value at point C
tinkle FMNIY fxlxti"
at
= - Ft÷l*÷i" tax
= fht "k ). €¥
n +1 ) !
:.
|Rnln)=Ht"a)(xaITtEiamfk : fkl : sin lm - ttxt XKIXHT ? + . .
.
Rn ( nl =fkt 'Yc ) ×K÷ th
.
( indeterminate,
but
fat"
lc )=± sink ) a task ) El
D R- k ) { ×h+÷
) !
And as n soo, Rube ) →o goes to zero for
any X,
so the
king Run = lnjdsoxanutf,
to } Took sense Hesin is tmefonox .
In '
ntfIi E¥i" IF iY÷'s
= t.tt# - Itt . ⇒+ .
.
- ÷
=/ , -1 , -
(omplei numbers
a
¥÷; a=Ywsoiisipo)
= reio
oaful for Multiplying or dividing complex numbers
Z ,= r ,
ei 01
Zz = rzei OzJ7z = r , rzeilotoz)
Eg= }et
.
.
zh = rhein 0
forri 6=(650+ is in o )
"= cos not is in no
singoso . i is in b )?_ 6520 +2 isinoaso - sing
=¢os2o. sink ) + i ( 2 sinooso ) = cos 20 + is in 20
z 'k=(reiofk= ikeiok= JT (wso=+ isino)
⇐ran ple
.
Costs
t.sn/%D25=CeiFf2Iei.t*irate 25
= en.
En= en
'
?it =A
. i = i times !
Tri chi USEFZHinstead of TiT = 27
.
÷ .
see
toys = ei¥eIei¥ ' =eie -
e
''T "
1
€-7 it full turn
÷oohfind roots of 25=8
z' 13
=p "3ei°13
.
.
*⇒*i .; :.mg?itisiaeIIMiiisr .
27134%to¥o minima¢3
"3=2
,Leith
,zeilmb? ¥2
,. itirs
,.
hits }