Instructors D. Krashen and P. Storm December 12, 2007

Math 241Final examination

Instructions. Answer the following problems carefully and completely. Show allyour work. Do not use a calculator. You may use both sides of one 8~ x 11 sheet ofpaper for handwritten notes you wrote yourself. Please turn in your sheet of noteswith your exam. There are 100 points possible. Good luck!

1

Name

Instructors's name

TA's name and time

1. (2)2. (14)3. (6)4. (2)5. (3)6. (8)7. (8)8. (5)9. (5)10. (10)11. (11)12. (6)13. (6)14. (14)

Total (100)

Here are some integrals you can use:

1. Write whether the following statement is true or false. (You do not need toshow any work.) The product of an odd function f with an odd function 9 isan odd function.

FALSE

2

2. Use a Fourier transform, a sine transform, or a cosine transform to find thedisplacement u(x, t), for x > 0 and t > 0, of a semi-infinite string if

u(O,t) = 0, u(x,O)= xe-x, and aau

I

= o.t t=o

You may assume the constant a2 of the wave equation is equal to 1. Your finalanswer may contain an integral. t

DO

#Ii W""i)'" ~LL(X,ih(){..:x) Jx ~ ~()

~ ~ 1 uLJ(rt)- 0

1!;1 ~~ ~ - o(J.ll(,{I-I:) t d u~i)_ d- -/ '\__ - ~ U.~o<)-t)

. { tY.-Z _ d~U1s if/;"-~ - ~i -- - ~;lu dU." ~ wM{vr ~c=w-~ fry

~ \ ;}d-[l- cX.1L&)-l:J;:: ~.

q 1[&'<1-1:) - A ClJ~d t) + 13 :5/i1(or'i)~ A ~ 6 w #cbsat ~ ~-

o ==1J 0~ = js { ~/H ~ = ~~ 0 .

o

x..{x

3

Scratch paper

=7 13 0::: 0 ~ 11 &)-t) "" A WS(d -/J

A ;:: 1JL~)O) ::: 'j-;;ilL{X)O)S:: 15ix;x~:: t::~)d-.

4' TT(o< ,-/; \ ::: 4 )~. UJs( rA1:).~ I) [1.+0( 00

~ uLx,-t) c::: ~~i U (,x)t)j = %J J-cX,,~ CftS&<{J sill (o(x) do( ·o

4

3. Find any two independent solutionsu(x, y) to the followingPDE:

fPu-=ufJxfJy

Neither of your solutions can be the zero function.

e.x+y.IS one S0\U+IO(1.

Olt1a+he\" so \u.b'cn .\3,

- x + Yfa-.::=: e. i:s () ot 0-- ton:s+a1'11: 'Pct .

9 -they P.rr2 fndependen-t .

4. Find a and b real numbers such that

10 - 5i = a + ib.

_:5 r..-Lt-~L

Ii -::.5/,+) b::::- ~ .

5

-

5. Let

Zl = 2cos('7!-jS)+ 2i sin(7!-jS)

Z2 = 4cos(37r/S) + 4i sin(37r/S)

Find a and b real numbers such that

Zl = a + ib.Z2

6

6. Show the complex function j(z) = z is not analytic at z = O.

NoY'4 let: ?,; = l X {I;'! y Ie-OLI.n. ~ -:- --LV~ ~ -.:: ~ J..::: J$ -r::: -1.~~ 0 y=70 iy y~~ Lj

1h'~ ~~ow:s. +he \i1Y),"t d()e...s: (101: eXI'~ -1:-

7

hrt \et =l::.X dZ X reCti.

-={ S;::k3-:=1-r. V\1'- x. x - .lO AO o

7. Find all points z in C satisfying the equation

sinz = 2.

Write the solutions in the form a + ib for a and b real numbers.

4-~ tL"i. ::: (e.i:j.)~- i( t~)~ II tZ=.t ~ ~Le - 1 =- ().

\ 11;..l:t 4-i.t(-1.<i>+Ltj - '). -+ 1'10

e. = - t?<~ - :a- LJl;Ad-..

~ ~ii d3 ~ t (J-:Hs)_

So oi1e set o~ sol'ns is 'i + a n - l b~(.Hf3){tjl vi A(1Y i(\'h2~e(".

i S;md tlr Iy J -13 > 0 ~ fhe othe, set of oS0/ ~5.~ iC~-(3') ( .r;;)

.

111 I ::'> J -I- ;)1f() - i% J -13 ~ () An;' It'!fecJer:8

L-et :t:::: ~ + lY . 1her\L~ -y ix · / .r;;\e ::: e. t:: :::: LCJ.i~3)

J.+E >0 ~ t1~( i Ld,-d3.J}'" ~ 4> x'" ~ + J1l"n

\i~ \ ;:: \iY \= e.-Y = Ii OA3) \ '" J.t f3- ~ y:c -j.J(J-+f3~

8. Compute the contour integral

where C is the circle Izi = 3.

-The. ~Lt. ~ ~ t is not AI1AlvHc. /1.t -tr.:lJ--_ 1f IT /

o

by t.u,J,/s ttJlh.

f-'~ ;;-j~:=O.C

f Q

9

9. Determine the pole(s) of 5 - 6/Z2. Find the order(s) of the pole(s). Computethe residue(s) at the pole(s).

lhe ~c.J . tli)::: 5 - CP/;gJ- IS

~ Lauren t se\\e.~ C~r1te\e do\dQ,\ ;) ~\th residue

~l-t) h~~ Y\~ o+her poles.

",I,euJr rj"HeY] if) the #0o" Jat 0) rJhere it has t?L poleofO.

10. Determine the pole(s) of1

1 - ez'

Find the order(s) of the pole(s). Compute the residue(s) at the pole(s).

{(~\'" l~e;' :fU\ is d.~i-pec;ojrc.) j,e. .f{;~tJrri)",f(2').

not Maly-lrc. Ai ;>=O. Let's ex.amine.fhj! ~i1:J\JJar.'ty Krst.

10

Scratch paper

2 1 J.. ~3

Consider- ~~k ~~ =: 1+ ~I+ %" + '-0 +.

By the ,,,-Ilo.fe,;t -j},is V0;1"-" ~~"e.s. o>~"~('~es. (JY\~iI of C.

:9 i'l.) i~ ArPirL tv,,-'y \'Jhe\e.. Seo)=: i .;:'1 1 / 3-.\ =: ~ / :!- \ ::: - i. L i)' /abJ)-J.. f\~l)

Jl ~ i-e) d~l ~l=l)) C LcJ ()

~ 1 l~ ):= 4 ({0):::: j.

it i-E: 0 a-..~.::o

If) By 4e~i J'jr~ \1

(A\\ simile

~ ~ is (!.Y14/11-lu,-- a;I:: 0-1-e I

vole at O. fhe resIdUe." cd; ()l~ I~ )

::: - -;-1 , \ ::: - 11.'00 li-e. ~ -l.=)()

J.-wi-pe.,' oJILi-l:y of ~['l))f-rf1 () AnY in-l-e~ e (" 1 ~nd

\il°-th resldue -1.

.IS

the o-the\ pCJles. are

the~ e. f 0/es ~r.e

11

11. Compute the integral

111" 1 dBo 5 + 4 cosB

e'fen S ()'Tr

1. r Je _- J. j ~+ 4c.o~.{:1-

-'11

r" J{ti J 5+J7\ Ji'-& --1:

1-:;::::;.JtL

12

12. Let C be the curve in the complex plane parametrized by C(t) = cos(t)+isin(t),for a :S t :S 71".(Note the 71"!)Compute the value of the contour integral

14

1 dzC z2

CLi) -::: CJ)si: tist'nt = e.lfo tr-

tl Li)::: ie. if'iT' "11

j h _ \ i it d-i: . f -it cd-,"?ie. :=L e tC

00

qJ

:= :z · L e-it] =: - i. (- i -1) = J '0

13. Consider the function

f(x) = edefined on the interval [0,2]. Let

for a ~ x ~ 1for 1 < x ~ 2

00

LBnsin (n;x)n=l

be a sine series for f(x). Using the same values for Bn, for all x in the real linedefine a function

00

g(x) =L Bn sin (n;x) .n=l

Find g( -5/2) and g( -5).

t8 \ ::: i.~

-fil) = - i (0 + i)~(S~) ~ tJtt +~) 0= 6(1) ~

ti-5J::- d(-1) == - 8(1)--to js. odd

15

14. Solve the Laplace equation Uxx+ Uyy = 0 for a function u(x, y) with 0 ::; x ::; 2,o ::; Y ::; 1 and boundary conditions:

ouu(O,y) = 0, ox(2,y) = 0, u(x,O)= 0,

(7rX

) . (57rX)u(x,l) = 3sin ttIf -2sm &\-*'1 .

uix,y) = ({x\. &Ly1

(IILx) CAy)+ (LX)' C/{y) '" 0q £B~-01ll:::-~

(Lx) &~ ) ,

('{x \+ /\ ~Lx)'" 0 ({()) = ~ '{J)-::: ()'A'> 0~'. Ld 0(= {0\.

(Lx)::: Ac.osCd.XI t 6 ;;1r1&XJ

~L()) -:; 0 q A ~ Q

( '{X)::: 0( 6 uS L.(' J,) ==C) q do(:::: 1+ 11"n ~~ . q IX == ~ + I n.. fIR rl-'" O} l,d) 3)'"

~ ~ 0". f=LX)::: A X-t 13 ~(o) = 0 ~ '~ :: 0i=='&)=: A ::: 0 -

'So tIP 'P 0 ;50 not Ml t1d renil 41j)e .16

Ii()\Jlated

0

0 -Lx

~<. 0 Scratch paper

~'. (LX) '" A Cj)$h(o(X! + g :s,nh{dX) & ct"'v.~Lo) ::: 0 -9 A ~ 0 ,

r::1{~)~ 150( cosh (;1.v{) = 0 q B:::: () .

~ (\0 el6eYJvalue:s~ 0,Sa +he. e.1~rJV ",Iue:s __ tlfe 1\ * (

'It + 1Q2) d-- #tD n:=: 11 1 I'") 3IJ /In 't :A U( v) ) 0\ I ) " ..

wiih e-r6enfd F)x):::: 130s'''cr;i;' x).

Il 9 G~ (y)::: ~o G.~) f GnLO\= 0

q ~)y1:::: fI Cns,nh{~· y) ,

~ uiAr):::: Ans,ohL~+ 1)y) Sin{{~+~)x) fdr n"'Q)j(dr..

~ uixr) '" 1 Unix}J 'i\-::: ()

Lt Lx) i) = 3 sin(1)- d slrJ( ~) ~ ~ An smh(L1 t ¥~ s,n(Lf+~) x)q b~r -the rL:::: 0 Mid IL"" 'J. c-1f ar~ niJYJze.r<J

A j A -~o ::: ~\) J-:::~)

<2 I(Jl} ) . J(jfXJ

'l . I(' 5 rr'11) .

C'.51rX)

uLx,v) :::: ~. sinn If "fO( iF - ~. I{S;

)

. sinh YJ SI(1 if '

! sfr1h{td ~mh Lf17

10...