Math Methods in Physics 1

Prof Wladwir lgraElp8 .

2016

Class # 4

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 }

y -6T : #-64 =

rei0r-64.o-Tlt2kTzll4-rH4ei0k.2vIeifFtKEH.ktqh2348zjTeiT82VTe@sfyeitlk.ei3tH.e,

's"K, .§ .

earth }a •

.

2uTe5Tl8zvpg.tt/8

V6T=±2±2i

.