1000 Solved Problems in Modern Physics

24 1 Mathematical Physics 1.25 Use the Beta functions to evaluate the definite integral ∫π/ 2 0 (cosθ) rdθ 1.26 Show that: (a)Γ( ...

1.2 Problems 25 1.2.6 Series............................................ 1.35 Find the interval of convergence for the series: x ...

26 1 Mathematical Physics 1.46 Show that: ∫ 4 2 2 x+ 4 x^2 − 4 x+ 8 dx=ln 2+π 1.47 Find the area included between the semi-cubic ...

1.2 Problems 27 1.56 Solve: d^3 y dx^3 − 3 d^2 y dx^2 + 4 y=0 (Osmania University) 1.57 Solve: d^4 y dx^4 − 4 d^3 y dx^3 + 10 d^ ...

28 1 Mathematical Physics 1.63 Solve: d^2 y dx^2 − 8 dy dx =− 16 y 1.64 Solve: x^2 dy dx +y(x+1)x= 9 x^2 1.65 Find the general s ...

1.2 Problems 29 1.71 Find a fundamental set of solutions to the third-order equation: d^3 y dx^3 − d^2 y dx^2 + dy dx −y= 0 1.2. ...

30 1 Mathematical Physics 1.76 The Bessel functionJn(x) is given by the series expansion Jn(x)= ∑ (−1)k(x/2)n+^2 k k!Γ(n+k+1) Sh ...

1.2 Problems 31 1.84 Evaluate ∮ c 4 z^2 − 3 z+ 1 (z−1)^3 dzwhenCis any simple closed curve enclosingz=1. 1.85 Locate in the fini ...

32 1 Mathematical Physics Fig. 1.5Soap film stretched between two parallel circular wires 1.2.13 StatisticalDistributions 1.93 P ...

1.3 Solutions 33 1.98 The alpha ray activity of a material is measured after equal successive inter- vals (hours), in terms of i ...

34 1 Mathematical Physics 1.2∇(xy^2 +xz)= ( ˆi∂ ∂x +ˆj ∂ ∂y +ˆk ∂ ∂z ) (xy^2 +xz) =(y^2 +z)ˆi+(2xy)ˆj+xkˆ = 2 ˆi− 2 ˆj−kˆ,at(− 1 ...

1.3 Solutions 35 (b) If the field is solenoidal, then,∇.rF(r)= 0 ∂(xF(r)) ∂x + ∂(yF(r)) ∂y + ∂(zF(r)) ∂z = 0 F+x ∂F ∂x +F+y ∂F ∂ ...

36 1 Mathematical Physics + ∫ 4 0 ( y^2 8 +y ) ydy 4 + ( y^2 8 −y ) dy (alongy^2 = 8 x) =+ 16 3 1.8 (a) It is sufficient to show ...

1.3 Solutions 37 Putx=Rcosθ,dx=−Rsinθdθ,y=Rsinθ,dy=Rcosθ,z= 0 , 0 < θ<∫ 2 π A.dr=− 2 R^2 ∫ sin^2 θdθ−R^2 ∫ cos^2 θdθ =− 2 ...

38 1 Mathematical Physics A unit vector normal to the surface is obtained by dividing the above vector by its magnitude. Hence t ...

1.3 Solutions 39 1.3.2 FourierSeriesandFourierTransforms................. 1.17 f(x)=^1 2 a 0 + ∑∞ n= 1 ( ancos (nπx L ) +bnsin ( ...

40 1 Mathematical Physics 1.18 The given function is of the square form. Asf(x) is defined in the interval (−π,π), the Fourier e ...

1.3 Solutions 41 which is consistent with Dirichlet’s theorem. Similar behavior is exhibited at x=π,± 2 π...Figure 1.8 shows fir ...

42 1 Mathematical Physics 1.21 Consider the Fourier integral theorem f(x)= 2 π ∫∞ 0 cosaxda ∫∞ 0 e−ucosaudu Putf(x)=e−x. Now the ...

1.3 Solutions 43 Thus the gamma function is an extension of the factorial function to numbers which are not integers. 1.24 B(m,n ...