1 Infinite Series, Power Series 2 Complex Numbers 3 Linear Algebra 4 Partial Differentitation 5 Multiple Integreals 6 Vector Analysis 7 Fourier Series And Transforms 8 Ordinary Differential Equations 9 Calculus Of Variations 10 Tensor Analysis 11 Special Functions 12 Series Solutions Of Differential Equations; Legendre, Bessel, Hermite, And Laguerre Functions 13 Partial Differential Equations 14 Functions Of A Complex Variable 15 Probability And Statistics expand_more
11.1 Introduction 11.2 The Factorial Function 11.3 Definition Of The Gamma Function; Recursion Relation 11.4 The Gamma Function Of Negative Numbers 11.5 Some Important Formulas Involving Gamma Functions 11.6 Beta Functions 11.7 Beta Functions In Terms Of Gamma Functions 11.8 The Simple Pendulum 11.9 The Error Function 11.10 Asymptotic Series 11.11 Stirling’s Formula 11.12 Elliptic Integrals And Functions 11.13 Miscellaneous Problems expand_more
Problem 1P: Expand the integrands of K and E [see ( 12.3 )] in power series in k2sin2 (assuming small k), and... Problem 2P: Use a graph of sin2 and the text discussion just before (12.4) to verify the equations (12.4). Note... Problem 3P: Computer plot graphs of K(k) and E(k) in (12.3) for k from 0 to $1 .$ Also plot 3D graphs of F(,k)... Problem 4P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 5P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 6P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 7P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 8P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 9P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 10P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 11P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 12P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 13P: In Problems 4 to 13, identify each of the integrals as an elliptic integral (see Examples 1 and 2 ).... Problem 14P: Find the circumference of the ellipse 4x2+9y2=36. Problem 15P: Find the length of arc of the ellipse x2+y2/4=1 between (0,2) and 12,3 (Note that here ba; see... Problem 16P: Find the are length of one arch of y=sinx. Problem 17P: Write the integral in equation (12.7) as an elliptic integral and show that (12.8) gives its value.... Problem 18P: Computer plot graphs of sn u, cn u, and dn u, for several values of k, say, for example,... Problem 19P: If u=ln(sec+tan), then is a function of u called the Gudermannian of u =gdu. Prove that:... Problem 20P: Show that for k=0:u=F(,0)=,snu=sinu,cnu=cosu,dnu=1 and for k=1:u=F(,1)=ln(sec+tan)or=gdu(Problem19),... Problem 21P: Show that the four answers given in Section 1 for 0/2d/cos are all correct. Hints: For the beta... Problem 22P: In the pendulum problem, =sing/lt is an approximate solution when the amplitude is small enough for... Problem 23P: A uniform solid sphere of density 12 is floating in water. (Compare Chapter 8, Problem 5.37. ) It is... Problem 24P: Sometimes you may find the notation F(,k) in (12.2) used when k1. Allowing this notation, show that... Problem 25P: As in Problem $24,$ show that 12Fsin1415,52=15Fsin123,25. format_list_bulleted