Complex analysis in a nutshell Definition A function f of one complex variable is said to be differentiable at z0 C if the limit f z f z0 lim z z0 z z0 exists and does not depend on the manner in which the variable z C approaches z0 Cauchy Riemann equations A function f z f x y u x y iv x y with u and v the real and the imaginary parts of f respectively is differentiable at z0 x0 iy0 if and only if it satisfies the Cauchy Riemann equations u v x v u v y x at x0 y0 Definition A function f is analytic at z0 if it is differentiable in a neighborhood of z0 Harmonic functions Let D be a region in IR2 identified with C A function u D IR is the real or imaginary part of an analytic function if and only if it is harmonic i e if it satisfies 2u 2u 0 x2 y 2 Cauchy formulas Let a function f be analytic in an open simply connected region D let be a simple closed curve contained entirely in D and traversed once counterclockwise and let z0 lie inside Then I f z dz 0 f z dz 2 if z0 z z0 I f z dz 2 i n f z0 n z z0 n 1 I n IN Definition A function analytic in C is called entire Zeros and poles If a function f analytic in a neighborhood of a point z0 vanishes at z0 and is not identically zero then f z z z0 k g z where k IN g is another function analytic in a neighborhood of z0 and g z0 6 0 In other words zeroes of analytic functions are always isolated and of finite order Rouche s theorem Suppose f and g are analytic in a disk z z0 r and g z f z on the circle z z0 r Then the functions f g and f have the same number of zeros counting multiplicities in z z z0 r Maximum principle If a function f is analytic in a disk z z0 r the f z max f whenever z z0 r z0 r 1 with equality if and only if f is a constant The maximum principle also holds for harmonic functions Liouville s theorem If a function f is entire and bounded then it is constant Singularities If a function f is differentiable in a punctured neighborhood of z0 but not at the point z0 itself then z0 is an isolated singularity of f There are three kinds of isolated singularities Removable singularities f z is bounded as z z0 Then f may be extended to a function analytic in a neighborhood of z0 Example f z sinz z z0 0 Poles There is a number k IN such that the function z z0 k f z has a removable singularity at z0 The smallest integer k with that property is called the order of the pole If k is the order of the pole then necessarily lim z z0 k f z 6 0 z z0 Essential singularities are those not of the two preceding kinds If z0 is an essential singularity of f then for any complex number w C a sequence zj converging to z0 can be found so that limj f zj w Taylor and Laurent series If a function f is analytic in a disk z z0 r then it expands into its Taylor series f z f z0 f 0 z0 z z0 f 00 z0 z z0 2 f n z0 z z0 n 2 The series is uniformly convergent in the disk z z0 r If a function f is analytic in an annular region r z z0 R then it expands into its Laurent series f z X cn z z0 n where cn n 1 2 i I f z dz n ZZ z z0 n 1 where is any curve lying in the annulus r z z0 R homeomorphic to a circle and traversed once in the counterclockwise direction Remark A function f has a pole of order k at z0 if and only if c k 6 0 cj 0 for all j k in its Laurent expansion centered at z0 An essential singularity gives rise to a Laurent series with an infinite negative part Definition The residue of f at z0 denoted by Resf z0 is the coefficient c 1 in its Laurent expansion centered at z0 or equivalently Resf z0 1 2 i 2 I f z dz where is a closed curve enclosing z0 and traversed once in the counterclockwise direction The residue theorem If a function f is analytic in a simply connected domain D except for a finite number of isolated singularities and if a curve is within D then I f z dz 2 i N X Resf zn n 1 where the zn s are the singularities of f contained within C The curve is as usual traversed once counterclockwise Finding residues Here are several formulas which simplify finding residues If f has a simple pole at z0 then Resf z0 lim z z0 f z z z0 If f has a pole of order m at z0 then Resf z0 z z lim 0 1 dm 1 z z0 m f z m 1 dz m 1 If f z g z h z where h has a simple zero at z0 and g is analytic at z0 then Resf z0 lim z z0 g z h0 z Using the residue theorem Integrals of the form Z 2 0 f sin cos d reduce to integrals along the unit circle using the substitution z ei d dz 1 1 1 1 cos z sin z iz 2 z 2i z Integrals along the entire real line Z f x dx may be often converted to limits of contour integrals The standard trick is to use one of the half circles CR Rei 0 or CR Rei 0 and the segment R R and then let R This requires that the integral of f along CR tend to zero Some integrals e g those containing hyperbolic functions require instead a rectangular contour with a segment parallel to R R chosen so as to produce a multiple of the original integral while integrating along the new segment R ic R ic Then one needs to make sure that contributions from the sides R R ic R R ic tend to zero Fancier contours may be required in some cases 3 Examples 1 Evaluate Z 2 0 i ee d 2 Prove that the polynomial p z z 47 z 23 2z 11 z 5 4z 2 1 has at least one root in the disk z 1 3 Suppose that f is analytic inside and on the unit circle z 1 and satisfies f z 1 for z 1 Show that the equation f z z 3 has exactly three solutions inside the unit circle 4 How many zeroes does the function f z 3z 100 ez have inside …