Mathematics - III

Code No: 07A3BS02 Set No. 1

II B.Tech I Semester Regular Examinations, November 2008MATHEMATICS-III

( Common to Electrical & Electronic Engineering, Electronics &Communication Engineering, Electronics & Instrumentation Engineering,

Electronics & Control Engineering, Electronics & Telematics, Electronics &Computer Engineering and Instrumentation & Control Engineering)

Time: 3 hours Max Marks: 80Answer any FIVE Questions

All Questions carry equal marks⋆ ⋆ ⋆ ⋆ ⋆

1. (a) Show that1∫

0

yq − 1 (log 1y)p−1 dy = Γ(p)

qp where p>0, q>0.

(b) Prove that β(m , 12) = 22m− 1 β(m , m ) [8+8]

2. (a) Prove that the function f(z) = u + i v , where f(z) = x3 (1+i)− y3 (1−i)x2 + y2 , z 6=

0 and f(0) = 0 is continuous and that Cauchy’s Riemann equations are satisfiedat the origin, yet f′(z) does not exist.

(b) Find the analytic function whose real part is y +ex cos y. [8+8]

3. (a) Prove that

i. ii = e−(4n+1)π

2

ii. log ii = −(2n + 12)π

(b) If tan( A + iB ) = x + iy , prove that x2 + y2 + 2x cot2A = 1. [8+8]

4. (a) Evaluate3+i∫

0

z2dz, along

i. the line y = x/3

ii. the parabola x =3y2

(b) Use Cauchy’s integral formula to evaluate∮

c

z3−2z+1

(z−i)2 dz, where c is the circle

|z| = 2. [10+6]

5. (a) Expand f(z) = 1+2zz2+z3 in a series of positive and negative powers of z.

(b) Expand ez as Taylor’s series about z = 1. [8+8]

6. Evaluate2π∫

0

dθ(5−3 sin θ)2

using residue theorem. [16]

7. Use Rouche’s theorem to show that the equation z5 + 15z + 1 = 0 has one root inthe disc |z| < 3

2and four roots in the annulus 3

2< |z| < 2 [16]

8. Show that the transformation w = z + 1z, converts that the radial lines θ= constant

in the z-plane in to a family of confocal hyperbolar in the w-plane. [16]

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1. (a) Evaluate

i.∞∫

0

√x . e−x3

dx

ii.∞∫

0

t−3/2 ( 1 − e− t ) dt

iii.1∫

0

x4 [ log ( 1x

)]3 dx

(b) Show thata∫

b

(x − b)m− 1 (a − x )n− 1 dx = (a − b)m+n−1 β(m , n) [12+4]

2. (a) Prove that the function f(z) =√

|xy| is not analytic at the origin even thoughthe C - R equations are satisfied thereat.

(b) Find the analytic function whose real part is y / (x2 + y2). [8+8]

3. (a) Separate into real and imaginary parts of cosh ( x + iy ) .

(b) Find all the roots of the equation

i. sin z = cosh 4

ii. sin z = i. [8+8]

4. (a) Prove that

i.∫

c

dzz−a

= 2πi

ii.∫

c

(z − a)ndz = 0, [n, any integer 6= -1]

(b) State and prove Cauchy’s integral theorem. [8+8]

5. (a) Give two Laurent’s series expansions in powers of z for f(z) = 1z2(1−z)

andspecify the regions in which these expansions are valued.

(b) Expand f(z) = 1z2

−3z+2in the region

i. 0 < |z − 1| < 1

ii. 0 < |z| < 2 [8+8]

6. (a) State and prove Cauchy’s Residue theorem.

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(b) Find the residue at z = 0 of the functionf(z) = 1+ez

sin z+z cos z[8+8]

7. State Rouche’s theorem. Prove that z7 - 5z3 + 12 = 0 all the roots of this equationlie between the circles |z| = 1 and |z| = 2 [16]

8. (a) Find and plot the image of the regions

i. x > 1

ii. y > 0

iii. 0 < y < 12

under the transformation w = 1z

(b) Prove that every bilinear transformation maps the totality of circle and straightlines in the z - plane on to the totality of circles and straight lines in the w-plane. [8+8]

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1. (a) Evaluate∞∫

0

x2 . e−x8

dx ×∞∫

0

x2 . e−x4

dx

(b) Show that

π/2∫

0

√cos θ dθ = 1

2Γ

(14

)Γ

(34

)

(c) Prove that1∫

0

x√(1−x5)

dx = 15β

(25, 1

2

)[6+5+5]

2. (a) Show that f(z) = xy2 (x+iy)x2 + y4 , z 6= 0 and f(0) = 0 is not analytic at z=0

although C- R equations are satisfied at the origin.

(b) If w = ϕ+iψ represents the complex potential for an electric field and ψ= 3x3y − y3

find ϕ. [8+8]

3. (a) Find the real part of the principal value of ilog(1+i)

(b) Separate into real and imaginary parts of sech ( x + i y ) . [8+8]


0

(x2 − iy)dz along the path

i. y = x

ii. y = x2

(b) Use Cauchy’s integral formula to evaluate∮

c

sin2 z

(z−π

6)3dz where c is the circle

|z| = 1 [8+8]

5. (a) Expand log (1 - z) when |z| < 1

(b) Determine the poles of the function

i. zcosz

ii. cot z. [8+8]

6. Show by the method of residues,π∫

0

dθa+b cos θ

= π√

a2−b2

(a > b > 0). [16]

7. (a) Apply Rouche’s theorem to determine the number of roots (zeros) ofP(z) = z4 - 5z + 1, with in annulus region 1 < |z| < 2.

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(b) Evaluate∮

C

f1(z)f(z)

dz where c is a simple closed curve, where f(z) = z2−1

(z2+z)2,

where c: |z| = 2 [16]

8. (a) Show that horizantal lines in z - plane are mapped to ellipser in w - planeunder the transformation w = sin z.

(b) Define Bilineer transformation. Determine the Bilinear transformation whichmaps z = 0, -i, 2i into w = 5i, ∞, −i

3[16]

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1. (a) Prove that Γ(m) Γ(m + 12

) =√

π

22m − 1 Γ(2m)

(b) Express the following integrals in terms of gamma function:

i.∞∫

0

xc

cx dx

ii.∞∫

0

a− bx2

dx [6+10]

2. The necessary and sufficient conditions for the function f(z) = u(x, y) + i v( x, y)to be analytic in the region R, are

(a) ∂u∂x

, ∂u∂y

, ∂v∂x

, ∂v∂y

are continuous functions of x and y in R.

(b) ∂u∂x

= ∂v∂y

, ∂u∂y

= − ∂v∂x

[16]

3. (a) Separate into real and imaginary parts of coth z

(b) If tan log ( x + i y) = a + i b where a2 + b2 6= 1, show thattan log ( x2 + y2 ) = 2a

i− a2−b2

[8+8]


0

(x2 + iy) dz along the path y = x and y = x2.

(b) Evaluate, using Cauchy’s integral formula∫

c

e2z

(z−1)(z−2)dz, where c is the circle

|z| = 3 [8+8]

5. (a) Expand f(z) = z−1z+1

in Taylor’s series about the point z = 0 and z = 1.

(b) Determine the poles of the function f(z) = 1−e2z

z4 [8+8]

6. (a) Determine the poles of the function f(z) = z2

(z+1)2(z+2)and the residues at each

pole.

(b) Evaluate∮

c

dx(z2+4)2

where c = |z − i| = 2 [8+8]

7. Show that the polynomial z5 + z3 + 2z + 3 has just one zero in the fist quadrantof the complex plane. [16]

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8. (a) Find the image of the infinite strip 0 < y < 12

under the transformationw = 1

z

(b) Show that the image of the hyperbola x2 - y2 = 1 under the transformationw = 1

zis the lemniscate p2 = cos 2 φ. [8+8]

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Code No: R05210201 Set No. 1

II B.Tech I Semester Supplimentary Examinations, November 2008MATHEMATICS-III





1. (a) Define Gamma function and evaluate Γ(1/2).

(b) Show that Γ(1/2)Γ(2n) = 22n−1 Γ(n) Γ(n+1/2).

(c) Define Beta function and show that β(m,n) = β(n,m). [6+6+4]

2. (a) Establish the formula P′

n+1 (x) − P′

n−1 (x) = (2n + 1) Pn (x).

(b) Prove that ddx

[x−n Jn(x)] = −x−nJn+1(x).

(c) When n is an integer, show that J−n(x)=(–1)nJn(x). [6+5+5]

3. (a) If f(z) is an analytic function, show that(

∂2

∂x2 + ∂2

∂y2

)|f(z)|2 = 4|f ′(z)|2.

(b) If tan log (x+iy) = a + i b where a2 + b2 6=1 prove that tanlog (x2+ y2) = 2a

1−a2−b2

. [8+8]

4. (a) Evaluate∫

c

(z2−2z−2) dz

(z2+1)2zwhere c is | z − i | = 1/2 using Cauchy’s integral for-

mula.

(b) Evaluate∫

C

(z2 + 3z + 2) dz where C is the arc of the cycloid x = a(θ + sin θ),

y = a (1 − cos θ) between the points (0,0) to (πa, 2a). [8+8]

5. (a) Expand logz by Taylor’s series about z=1.

(b) Expand 1(z2 +1)(z2+2)

in positive and negative powers of z if 1 < |z| <√

2. [8+8]

6. (a) Find the poles and residues at each pole of the function zez

(z−1)3.

(b) Evaluate∫

C

2ezdzz(z−3)

where C is |z| = 2 by residue theorem. [8+8]

7. (a) Evaluate∫ 2π

0Cos2θ

5+4Cosθdθ using residue theorem.

(b) Evaluate∫∞

−∞

x2dx(x2+1)(x2+4)

using residue theorem. [8+8]

8. (a) Find and plot the map of rectangular region 0≤x≤1; 0≤y≤2, under the trans-formation w =

√2 eiπ/4z +(1-2i).

(b) Find the bilinear transformation that maps the points 0,i,1 into the points–1,0,1. [8+8]

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1. (a) Prove that Γ(n) =1∫

0

[log . 1x]n−1dx

(b) Prove that∞∫

0

x8(1−x6)dx

(1+x)24= 0 using β − Γ functions. [8+8]

2. (a) When n is a positive integer show that Jn(x) = 1π

π∫

0

cos(nθ − x sin θ)dθ.

(b) Show that x3 = 25P3 (x) + 3

5P1(x). [8+8]

3. (a) Define analyticity of a complex function at a point P and in a domain D.Prove that the real and imaginary parts of an analytic function satisfy Cauchy? Riemann Equations.

(b) Show that the function defined by f(z) = x3(1+i)−y3(1−i)x2+y2 at z 6= 0 and f(0) = 0

is continuous and satisfies C-R equations at the origin but f ′(0) does not exist.[8+8]

4. (a) Evaluate∫

C

ez dz(z2+π2)2

where C is | z | = 4 using Cauchy’s integral formula.

(b) Evaluate∫

C

dzz3(z+4)

where C is | z | = 2 using Cauchy’s integral formula. [8+8]

5. (a) State and prove Taylor’s theorem.

(b) Find the Laurent series expansion of the functionz2

−6z−1(z−1)(z−3)(z+2)

in the region 3< |z+2| <5. [8+8]

6. (a) Find the poles and residues at each pole f(z) = zez

(z+2)4(z−1)where z=-2 is a

pole of order 4.

(b) Find the poles and the residues at each pole of z(z2

−4).

(c) Evaluate∫

C

e2z

(z+1)3dz using residue theorem where c is [z] = 2. [5+5+6]

7. (a) Use Rouche’s theorem to show that the equation z5 + 15 z + 1=0 has oneroot in the disc |z| < 3

2and four roots in the annulus 3

2< |z| < 2.

(b) Evaluate∞∫

−∞

cos xdx(a2+x2)

(a > 0) using residue theorem. [8+8]

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8. (a) Show that the image of the hyperbola x2–y2=1 under the transformationw=1/z is r2= cos 2θ.

(b) Show that the transformation u = 2z+3z−4

changes the circle x2 + y2 –4x = 0into the straight line 4u+3=0. [8+8]

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1. (a) Show that1∫

0

xm (log x)ndx = (−1)nn!

(m+1)n+1 where n is a positive interger and m>-1.

(b) Show that β(m,n)=∞∫

0

yn−1

(1+y)m+n dy.

(c) Show that∞∫

0

x4e−x2

dx = 3√

π

8. [6+5+5]

2. (a) When n is a positive integer show that Jn(x) = 1π

π∫

0

cos(nθ − x sin θ)dθ.

(b) Show that x3 = 25P3 (x) + 3

5P1(x). [8+8]

3. (a) Define analyticity of a complex function at a point P and in a domain D.Prove that the real and imaginary parts of an analytic function satisfy Cauchy? Riemann Equations.

(b) Show that the function defined by f(z) = x3(1+i)−y3(1−i)x2+y2 at z 6= 0 and f(0) = 0

is continuous and satisfies C-R equations at the origin but f ′(0) does not exist.[8+8]

4. (a) Evaluate∫

C

(sin πz2+cos πz2) dz

(z−1)(z−2)where C is the circle |z| = 3 using Cauchy’s integral

formula.

(b) Evaluatez=1+i∫

z=0

(x2 + 2xy + i(y2 − x))dz along y=x2. [8+8]

5. (a) State and derive Laurent’s series for an analytic function f (z).

(b) Expand 1(z2

−3z+2)in the region.

i. 0 < | z – 1 | < 1

ii. 1 < | z | < 2. [8+8]

6. (a) Find the residue of f(z) = Z2−2Z

(Z+1)2(Z2+1)at each pole.

(b) Evaluate∮

c

4−3zz(z−1)(z−2)

dz where c is the circle | z | = 32

using residue theorem.

[8+8]

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7. (a) Evaluate2π∫

0

dθa+b cos θ

, a>0, b>0 using residue theorem.

(b) Evaluate∞∫

0

dx(1+x2)2


8. (a) Define conformal mapping. Let f(z) be an analytic function of z in a domainD of the z-plane and let f ′ (z) 6=0 in D. Then show that w=f(z) is a conformalmapping at all points of D.

(b) Find the bilinear transformation which maps the points (–i, o, i) into the point(–l, i, l) respectively. [8+8]

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1. Evaluate using β − Γ functions.

(a)1∫

0

x2(log 1x)3dx

(b)π/2∫

0

sin7/2 θ cos3/2 θ dθ

(c) Show that1∫

−1

(1 + x)m−1(1 − x)n−1dx = 2m+n+1β (m, m). [5+5+6]

2. (a) Prove that ddx

(xJn Jn+1) = x(J2n − J2

n+1).

(b) Express x3+2x2–x–3 in terms of Legende polynomials. [8+8]

3. (a) If f(z) = u + iv is an analytic function and u–v = cos x+sin x−e−y

2 cos x−ey−e−y , find f(z)

subject to the condition f(π/2) = 0.

(b) Separate the real and imaginary parts of logsinz. [8+8]

4. (a) Evaluate∫

C

(z3−sin 3z) dz

(z−π

2)3

withC: | z | = 2 using Cauchy’s integral formula.

(b) Evaluate(1,1)∫

(0,0)

(3x2 + 4xy + ix2) dz along y=x2.

(c) Evaluate∫

C

dzez(z−1)3

where C: | z | = 2 Using Caucy’s integral theorem. [5+5+6]

5. (a) Find the Laurent series expansion of the function z2−1

z2+5z+6about z = 0 in the

region 2 < |z| < 3.

(b) Evaluate f(z) = 2(2z+1)3

about (i) z = 0 (ii) z = 2. [8+8]

6. (a) Find the poles and residue at each pole of the function z2

(z4−1)

.

(b) Evaluate∫

C

(1+ez) dz

(z cos z+sin z)where C is |z| = 1 by residue theorem. [8+8]

7. (a) Show thatπ∫

0

Cos2θ1−2aCosθ+a2 = πa2

√

1−a2, (a2 < 1) using residue theorem.

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(b) Show by the method of contour integration that∞∫

0

Cosmx(a2+x2)2

dx = π4a3 (1 + ma)e−ma,

( a > 0 , b > 0 ). [8+8]

8. (a) Show that under the transformation w = (z-i)/ (z+i), real axis in the z-planeis mapped into the circle | w | = 1. Which portion of the z-plane correspondsto the interior of the circle?

(b) Prove that the transformation w= sin z maps the families of lines x=a andx=b into two families of confocal central conics. [8+8]

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II B.Tech I Semester Supplimentary Examinations, February 2008MATHEMATICS-III





1. (a) Evaluate1∫

0

x4(log 1

x

)3dx using β − Γ functions.

(b) Evaluate∞∫

0

xdx(1+x6)

using β − Γ functions.

(c) Evaluate−1∫

0

x4√

a2 − x2 dx using β − Γ functions. [5+6+5]

2. (a) Show that J4(x)=(

48x3 −

8x

)J1(x) +

(1 − 24

x2

)J0(x).

(b)1∫

−1

(1 − x2) (P ′

n)2dx = 2n(n+1)

(2n+1). [8+8]

3. (a) If w = u + iv is an analytic function of z and u + v = sin 2xcos h 2y − cos 2x

then find

f(z).

(b) If sin (θ + iα) = cos α + i sin α, then prove that cos2θ =sin2α. [8+8]

4. (a) Find f(z) and f(3) if f(a)=∫

C

(2z2−z−2) dz

(z−a)where C is the circle |z| = 2.5 using

Cauchy’s integral formula.

(b) Evaluate∫

C

logzdz where C is the circle |z| = 1 using Cauchy’s integral for-

mula.[8+8]

5. (a) Find the Laurent expansion of 1(z2

−4z+3), for 1 < | z | < 3.

(b) Expand the Laurent series of z2−1

(z+2)(z+3), for | z | > 3. [8+8]

6. (a) Find the poles and the residues at each pole of f(z)= zz2+1

.

(b) Evaluate∫

C

zezdz(z2+9)

where c is |z | = 5 by residue theorem. [8+8]

7. (a) Show thatπ∫

0

Cos2θ1−2aCosθ+a2 = πa2

√

1−a2, (a2 < 1) using residue theorem.

(b) Show by the method of contour integration that∞∫

0

Cosmx(a2+x2)2

dx = π4a3 (1 + ma)e−ma,

( a > 0 , b > 0 ). [8+8]

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8. (a) Show that the image of the hyperbola x2–y2=1 under the transformationw=1/z is r2= cos 2θ.

(b) Show that the transformation u = 2z+3z−4

changes the circle x2 + y2 –4x = 0into the straight line 4u+3=0. [8+8]

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1. (a) Show that β(m,n) = (Γ (m) Γ (n))/Γ (m + n).

(b) Show that1∫

0

xn

√

1−x2dx = 2.4.6....(n−1)

1.3.5....nwhere n is an odd integer.

(c) Show thatπ/2∫

0

√tan θ dθ = Γ(1/4)Γ(3/4)

2. [6+5+5]

2. (a) Establish the formula P′

n+1 (x) − P′

n−1 (x) = (2n + 1) Pn (x).

(b) Prove that ddx

[x−n Jn(x)] = −x−nJn+1(x).

(c) When n is an integer, show that J−n(x)=(–1)nJn(x). [6+5+5]

3. (a) Find the analytic function whose imaginary part isf(x,y) = x3y – xy3 + xy +x +y where z = x+iy.

(b) Prove that(

∂2

∂x2 + ∂2

∂y2

)|Real f(z)|2 = 2|f ′(z)|2 where w =f(z) is analytic.

[8+8]

4. (a) Find f(z) and f(3) if f(a)=∫

C

(2z2−z−2) dz

(z−a)where C is the circle |z| = 2.5 using

Cauchy’s integral formula.

(b) Evaluate∫

C

logzdz where C is the circle |z| = 1 using Cauchy’s integral for-

mula.[8+8]

5. (a) State and prove Laurent’s theorem.

(b) Obtain all the Laurent series of the function 7z−2(z+1)(z)(z−2)

about z= -2. [8+8]

6. (a) Find the poles and the residue at each pole of f (z) = sin2 z

(z−π/6)2

(b) Find the poles and the residue at each pole of f (z) = zez

(z−1)3. [5+5+6]

(c) Evaluate∫

C

cos π z 2dz(z−1)(z−2)

where C is |z | = 3/2. [5+5+6]

7. (a) Show that2π∫

0

dθa+bsinθ

= 2π√

a2−b2

, a > b> 0 using residue theorem.

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(b) Evaluate by contour integration∞∫

0

dx(1+x2)

. [8+8]

8. (a) Show that the transformation w= i(1–z) / (i–z), maps the interior of the circle|z|=1 in to the upper half of the w-plane, the upper semi circle into positivehalf of real axis and lower semi circle into negative half of the real axis.

(b) By the transformation w = z2 show that the circle | z–a | = c (a and c arereal) in the z plane correspond to the limacons in the w-plane. [8+8]

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1. (a) Show that β(m,n)=2π/2∫

o

sin2m−1θ cos2n−1 θdθ and deduce that

π/2∫

o

sinnθ dθ =π/2∫

o

cosnθ dθ =1

2Γ(n+1

2)Γ( 1

2)

Γ(n+2

2)

.

(b) Prove that Γ(n) Γ( (1-n)= πsin nπ

.

(c) Show that∞∫

0

xme−a2x2

dx = 12am+1 Γ

(m+1

2

)and hence deduce that

∞∫

o

cos(x2)dx =∞∫

o

sin(x2)dx = 1/2√

π/2. [5+5+6]

2. Prove that1∫

−1

Pm(x)Pn(x)dx =

{0 if m 6= n

22n+1

if m = n. [16]

3. (a) Find the analytic function whose imaginary part isf(x,y) = x3y – xy3 + xy +x +y where z = x+iy.

(b) Prove that(

∂2

∂x2 + ∂2

∂y2

)|Real f(z)|2 = 2|f ′(z)|2 where w =f(z) is analytic.

[8+8]

4. (a) Evaluate∫

C

(z3−sin 3z) dz

(z−π

2)3

withC: | z | = 2 using Cauchy’s integral formula.

(b) Evaluate(1,1)∫

(0,0)

(3x2 + 4xy + ix2) dz along y=x2.

(c) Evaluate∫

C

dzez(z−1)3

where C: | z | = 2 Using Caucy’s integral theorem. [5+5+6]

5. (a) Expand f(z) = e2z

(z−1)3about z=1 as a Laurent series. Also find the region of

convergence.

(b) Find the Taylor series for zz+2

about z=1, also find the region of convergence.[8+8]

6. (a) Find the poles and the residues at each pole of f(z)= zz2+1

.

(b) Evaluate∫

C

zezdz(z2+9)

where c is |z | = 5 by residue theorem. [8+8]

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7. (a) Evaluate∫ 2π

0Cos2θ

5+4Cosθdθ using residue theorem.

(b) Evaluate∫∞

−∞

x2dx(x2+1)(x2+4)


8. (a) Find the image of the infinite strip 0<y<1/2 under the transformation w=1/z.

(b) Find the bilinear transformation which maps the points (–1, 0, 1) into thepoints (0, i, 3i). [8+8]

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1. (a) Evaluateπ/2∫

0

sin2 θ cos4 θ dθ = π32

using β − Γ functions.

(b) Prove that∞∫

0

√x e−x2

dx = 2∞∫

0

x2e−x4

dx using β − Γ functions and evaluate.

(c) Show that∞∫

0

xm−1

(x+a)m+n dx = a−nβ(m, n). [5+6+5]

2. (a) Show that the coefficient of tn in the power series expansion of ex

2(t− 1

t)is Jn(x).

(b) Prove that1∫

−1

xPn(x)Pn−1(x) = 2n(4n2

−1). [8+8]

3. (a) Determine the analytic function f(z) =u+iv given that3u + 2v = y2 – x 2 + 16x.

(b) If sin (α + iβ) = x + iy then prove that x2

cosh2 β+ y2

sinh2 β= 1 and x2

sin2 α− y2

cos2 α= 1.

[8+8]

4. (a) Evaluate∫

c

log z dz

(z−1)3where c: |z − 1| = 1

2, using Caucy’s integral Formula.

(b) State and prove Cauchy’s Theorem. [8+8]

5. (a) Expand logz by Taylor’s series about z=1.

(b) Expand 1(z2 +1)(z2+2)

in positive and negative powers of z if 1 < |z| <√

2. [8+8]

6. (a) Determine the poles of the function and the corresponding residues of f(z)=z+1

z2(z−2).

(b) Evaluate∮

Cdz

sinhz, where c is the circle | z | = 4 using residue theorem. [8+8]

7. (a) Evaluate by residue theorem2π∫

0

dθ(2+cosθ)

.

(b) Use the method of contour integration to evaluate∞∫

−∞

x2dx(x2+a2)3

. [8+8]

8. (a) Find the image of the domain in the z-plane to the left of the line x=–3 underthe transformation w=z2.

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(b) Find the bilinear transformation which transforms the points z=2,1,0 intow=1,0,i respectively. [8+8]

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Mathematics - III