By Siemons J.
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3), we can easily deduce the properties of the logarithmic integral from those of the exponential integral. 6) implies the expansion where z belongs to the plane with cuts along the segments [ – ∞, 0] and [1, ∞]. 7 Finally, we note that the results of Sec. 2 imply the asymptotic representation where |rn(z)| = 0(|log z|−n−1) for large values of |log z|. In particular, for |z| < 1, and in this case the sector is just |arg z| π – δ. 5. Application to Electromagnetic Theory. Radiation of a Linear Half-Wave Oscillator8 As a simple example of the application of the special functions studied in this chapter, we consider the electromagnetic energy radiated by a linear oscillator of length 2l = λ/2, driven by an alternating current I of frequency ω = 2πc/λ (c is the velocity of light and λ the wavelength), whose distribution along the conductor is give by (see Figure 10).
Prove that for real y. 2. 6), verify the identity 3. Prove that 4. Verify the formula 5. Derive the formula Hint. Calculate the logarithmic derivatives of both sides of (i). 6. Derive the following integral representation of the square of the gamma function, where K0(t) is Macdonald’s function (defined in Sec. 7): Hint. 1) and the integral in Problem 2. 7. Derive the asymptotic formulas where α and β are arbitrary constants, and |arg z| π – δ. Hint. Use the results of Sec. 4. 8. Derive the asymptotic formula where as |t| → ∞, r(x, y) → 0 uniformly in the strip |x| α (α is a constant).
I. Markushevich, Theory of Functions of a Complex Variable, Vol. I (translated by R. A. J. 6, p. 326. 6 According to this principle, which we will use repeatedly, if f(z) and φ(z) are analytic in a domain D and if f(z) = φ(z) for all z in a smaller domain D* contained in D, then f(z) = φ(z) for all z in D. The same is true if f(z) = φ(z) for all z in any set of points of D with a limit point in D, say, a line segment. See A. I. Markushevich, op. 1, p. 369. 7 For the evaluation of the integral in the last step, see E.
2-Designs and a differential equation by Siemons J.