By Yu. V., M. V. Fedoryuk, and M. I. Shabunin Sidorov
This booklet relies on greater than ten years event in instructing the speculation
of capabilities of a fancy variable on the Moscow Physics and expertise
Institute. it's a textbook for college students of universities and institutes
of know-how with a sophisticated mathematical program.
We think that it could even be used for self sustaining study.
We have under pressure the equipment of the speculation which are frequently utilized in utilized
sciences. those equipment comprise sequence expansions, conformal mapping,
software of the speculation of residues to comparing certain integrals,
and asymptotic tools. the cloth is based in a manner that might
supply the reader the utmost guidance in getting to know the fundamentals of the
concept. To this finish we have now supplied a variety of worked-out
examples. we are hoping that those might help the reader collect a deeper figuring out of the speculation and adventure in challenge solving.
The e-book was once translated from Russian through Eugene Yankovsky and was once first released via Mir in 1985.
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Extra info for Lectures on the Theory of Functions of a Complex Variable
2), the function fP = arg z defined above in terms of inverse trigonometric functions: if x > 0, arg z = arc tan (ylx) arg z = n + arg tan (ylx) if x < 0 and y > 0, arg z = -:rt + arc tan (ylx) if x < 0 and y < 0. However, in the case of an arbitrary domain there is no simple Fig. 26 Fig. 27 formula that relates a branch of· arg z and inverse trigonometric functions, since these functions vary either from -n/2 to n/2 or from 0 to n, while the function arg z may vary within any limits. A more convenient representation of arg z is the integral representation (see Eq.
Lim f (zn) = A. For brevity we will often write Zn n-oo lim f (z) instead of lim f (z). Z-+a, z EE z-a Theorem 1 of Sec. 2 implies that the existence of the limit lim f (z), with f (z) = u (x, y) iv (x, y) and a = a + i~, is + z ... a equivalent to the existence of two limits, lim u (x, y) and x-a y-(3 lim v (x, y), with x ... a y ... (3 lim f (z) =lim u (x, y) -r i lim v (x, y). Limits of functions of a complex variable have the same properties as limits of functions of a real variable, namely, if lim f (z) =A and lim g (z) = B, z ...
E. any two points belonging to D can be connected by a curve (perhaps unlimited) whose all points belong to D. A boundary point of D is a point in any neighborhood of which there are points belonging to D and points not belonging to D. The set of the boundary points of a domain is the boundary of this domain. Joining a domain D and its boundary points results in the closure -of D denoted by D. In what follows we will only consider domains whose boundaries -consist of a finite number of piecewise smooth curves and isolated points.