Evaluation of four irrational cosine definite integrals using residue theory
For the four integrals \[ D_ j(a)=\int^{2\pi}_{0}(a+\cos^ j\theta)^{-1/2} d\theta,\quad N_ j(a)=\int^{2\pi}_{0}(a+\cos^ j\theta)^{1/2} d\theta,\quad j=1,2, \] Laurent series, convergent for sufficiently large values of \(| a|\), are ``derived'' by rather dubious calculations with residues which, miraculously, lead to correct results (although written in a clumsy way). If the authors were familiar with the concept of multi-valued functions, branch points and branch cuts they easily could have established their series in a clean way. - On top of this shortcoming the paper is completely unnecessary since the four integrals are essentially the well known complete elliptic integrals. By the substitution \(\theta =2\phi\) in \(N_ 1\), \(D_ 1\) and trivial operations one obtains with \(k:=(a+1)^{-1/2}:\) \[ D_ 1=4k\cdot K(k\sqrt{2}),\quad N_ 1=(4/k)E(k\sqrt{2}),\quad D_ 2=4k\cdot K(k),\quad N_ 2=(4/k)E(k), \] where \[ K(k):=\int^{\pi /2}_{0}(1- k^ 2 \sin^ 2\phi)^{-1/2} d\phi,\quad E(k):=\int^{\pi /2}_{0}(1- k^ 2 \sin^ 2\phi)^{1/2} d\phi \] are the complete elliptic integrals of first and second kind, respectively. These functions have been well investigated centuries ago; formulas may be found, e.g., in \textit{M. Abramowitz} and \textit{I. A. Stegun} [(*) Handbook of mathematical functions with formulas, graphs, and mathematical tables (1964; Zbl 0171.385), p. 589-600]. K(k) and E(k) may be evaluated efficiently (more efficiently than by a series) by the arithmetic-geometric mean [(*), p. 598]. Series for small values of k are \[ K(k)=\pi /2\cdot_ 2F_ 1(1/2,1/2;1;k^ 2),\quad E(k)=\pi /2\cdot_ 2F_ 1(-1/2,1/2;1;k^ 2) \] [(*), Equation 17.3.9-12], where \[ _ 2F_ 1(a,b;c;z)=1+(ab/c\cdot 1!)z+(a(a+1)b(b+1))/(c(c+1)\cdot 2!)z^ 2+... \] is the Gauss hypergeometric series [(*), p. 556ff]. Kummer's quadratic transformations, e.g. [(*), Equation 15.3.16], yield the authors' series, e.g. \[ D_ 1(a)=2\pi /\sqrt{a}\cdot_ 2F_ 1(1/4,3/4;1;a^{-2}). \]
scientific article; zbMATH DE number 4126556