Begin2.DVI

Recall Euler’s infinite product formula for sin θ(see Example 4-38) sin(πx ) = πx ( 1 −x 2 12 )( 1 −x 2 22 )( 1 −x 2 32 ) ··· (1 ...

Now interchange the roles of summation and integration on the right hand side of equation (12.119) to obtain ζ(z) = ∑∞ r=1 1 rz ...

can be written as a product of factors having the form x^2 n− 2 xncos nθ + 1 = 2 n−^1 n∏− 1 r=0 (x^2 − 2 xcos(θ+ 2 rπ n ) + 1) ( ...

are given by x^2 − 2 xcos(θ+^2 rπ n )+1 for r= 0, 1 , 2 ,.. ., n − 1. This implies x^2 n− 2 xncos nθ +1 can be expressed as x^2 ...

Derivatives of ln Γ( z) Using the Weierstrass definition of the Gamma function 1 Γ(z)=ze γz ∏∞ n=1 [( 1 + 1 z ) e−z/n ] (12.135) ...

is defined as the Hurwitz^4 Zeta function and satisfies ζ(n,0) = ζ(n),the Zeta function. The function ψn(z) = (−1)n+1n!ζ(n+ 1 , ...

Taylor series expansion for ln Γ( x+ 1) Make reference to the equations (12.136), (12.137), (12.138), (12.139), and verify that ...

The product nm is used in the definition of Γ(nz ) to show that the equation (12.146) simplifies after a lot of careful algebra ...

Using the results from pages 361-362, one can write ∑n i=1 ∆Ψ(a+i) = ∑n i=1 1 a+i= Ψ(a+n+ 1) −Ψ(a+ 1) (12.153) As an example of ...

where c 0 , c 1 , c 2 , c 3 ,... are constants to be determined. Substitute the representations (12.157) and (12.158) into the d ...

Example 12-10. Determination of series Find a power series expansion to represent the function y=y(x) = (h+x)n, where his a cons ...

Here the recurrence formula is (m+ 1)h cm+1 +m cm=n cm or cm+1 = (n−m) (m+ 1)h cm (12.171) for m= 0 , 1 , 2 ,.. .. If y=y(x) = ( ...

Figure 12-8. The Laplace transform operator. In the defining equation (12.175) the parameter sis selected such that the in- tegr ...

Example 12-12. Laplace transform Find the Laplace transform of eαt Solution By definition L { eαt ; t→s } = ∫∞ 0 eαte−st dt = ∫∞ ...

Inverse Laplace Transformation L−^1 The symbol L−^1 is used to denote the inverse Laplace transform operator with the property t ...

Figure 12-9. Inverting the Laplace transform. Properties of the Laplace transform Using the definition of the Laplace transform ...

Example 12-14. First shift property If F(s) = L{f(t)}, show that L { eatf(t) } =F(s−a)or eatf(t) = L−^1 {F(s−a)} Solution By def ...

Properties of the Laplace Transform Function f(t) Laplace Transform F(s) Comment c 1 f(t) c 1 F(s) linearity property f′(t) sF ( ...

Note the following repetitive properties exhibited by the above table. (i) The derivative property, when expressed in words stat ...

so that the differential equation in the t-domain becomes an algebraic equation in the s-domain. The resulting algebraic equatio ...