Calculus and Complex Variables

Cauchy integral theorem

Cauchy's integral theorem states that if f is analytic in a simply connected domain D, then \oint_C f(z)\,dz = 0 for every simple closed contour C lying in D. The hypothesis concerns the whole region enclosed by C, not only the path itself, so a singularity inside the contour blocks the direct use of the theorem. For example, \oint_{|z|=2} \frac{dz}{(z-4)^2} = 0 because the double pole at z=4 lies outside the contour, and the integrand is analytic on and inside it.

Analyticity is a sufficient condition for a zero integral, not a necessary one. The function 1/z^2 is not analytic at z=0, yet its integral over the unit circle vanishes because its residue at 0 is zero. Thus the theorem cannot be applied directly, but the value can still be zero. Every polynomial is entire, so its integral over any closed path is zero.

Global extrema of a function

A global extremum of a function on its domain is found by comparing candidate values. The candidates are the critical points where f'(x)=0 or f' fails to exist, together with the limiting behaviour of f as x\to\pm\infty when the domain is unbounded. A sign analysis of f' tells whether a critical point is a maximum or a minimum. For example, f(x)=x^2e^{-x} on [0,\infty) has f'(x)=x(2-x)e^{-x}, so the critical points are 0 and 2. Since f(0)=0 and f\to 0 as x\to\infty, the minimum value is 0 at x=0.

For a pointwise maximum of two functions, the graph is piecewise and follows whichever branch is larger. Its minimum occurs either at a crossing point of the two branches or at a turning point of one branch. For instance, \max\{x^2,\,2-x\} switches branches at x=1 and x=-2, and its smallest value is 1, attained at x=1.

Limit of recursive sequence

A recursively defined sequence x_n = a\,x_{n-1} + b with |a|<1 converges, and its limit L must be a fixed point of the recursion, so L = aL + b. This gives L = \dfrac{b}{1-a}. The limit does not depend on x_0 when |a|<1, because the error x_n - L is multiplied by a at every step. For example, x_n = \tfrac{1}{3}x_{n-1} + 2 has limit 3.

When |a|\ge 1, the error does not shrink, so the sequence may diverge or oscillate even though a fixed point exists. The fixed-point equation gives only a candidate limit, and the contraction condition must be checked before the candidate is accepted.

Extrema on closed interval

A continuous function on a closed bounded interval [a,b] attains both a maximum and a minimum by the extreme value theorem. The candidates are the critical points inside (a,b) together with the two endpoints. The largest and smallest values among these candidates are the global extrema on the interval. For example, for f(x)=x^3-3x on [-2,2], the critical points are x=\pm1. Comparing f(-2)=-2, f(-1)=2, f(1)=-2, and f(2)=2 gives the minimum -2.

If the vertex of a quadratic lies outside the interval, the function is monotone on that interval. The extremum then lies at an endpoint, and the interior critical point can be ignored.

Laurent series expansion

A Laurent series represents f(z) in an annulus r<|z-z_0|<R as f(z)=\sum_{n=-\infty}^{\infty} a_n (z-z_0)^n. The terms with negative powers form the principal part, and the coefficient a_{-1} of (z-z_0)^{-1} is the residue at z_0. The expansion depends on the annulus chosen, so the same function can have different Laurent series in different regions.

The practical method is to rewrite f as a combination of simple terms and expand each one with a geometric series. For |u|<1, \dfrac{1}{1-u}=\sum_{n\ge 0}u^n, and for |u|>1, \dfrac{1}{1-u}=-\sum_{n\ge 0}u^{-n-1}. For example, \dfrac{1}{z(z-2)}=\dfrac12\Big(\dfrac{1}{z-2}-\dfrac1z\Big). In 0<|z|<2, the term \dfrac{1}{z-2} becomes -\sum_{n\ge 0} \dfrac{z^n}{2^{n+1}}, so the coefficient of z^{-1} is -\tfrac12.

Complex number modulus and argument

For a complex number z=x+jy, the modulus is |z|=\sqrt{x^2+y^2}. The argument is \arg z=\tan^{-1}(y/x), with the quadrant fixed by the signs of x and y. When z is a quotient w/\bar w of a number and its conjugate, |z|=1 because numerator and denominator have equal moduli. Also \arg z = 2\arg w, since \arg \bar w = -\arg w. For example, z=\dfrac{1+j}{1-j}=j has modulus 1 and argument \pi/2 = 2\cdot\pi/4.

Be careful not to confuse 2\tan^{-1}(b/a) with \tan^{-1}(2b/a). The doubling acts on the argument itself, not on the ratio inside the arctangent.

Power series expansion around a point

A function expanded as a power series about z_0 has the form \sum_{k\ge 0} a_k (z-z_0)^k, and its coefficients satisfy a_k = \dfrac{f^{(k)}(z_0)}{k!}. The radius of convergence equals the distance from z_0 to the nearest singularity of f, so the region of convergence is a disc about z_0.

In practice, rewrite the denominator in terms of (z-z_0) and apply the geometric series. For example, \dfrac{1}{2-z} about z=1 becomes \dfrac{1}{1-(z-1)}=\sum_{k\ge 0}(z-1)^k, with radius 1. The constant term a_0=1 agrees with the value f(1)=1, which is a quick check that catches sign errors.

Limits of elementary functions

Limits of products such as x\sin(1/x) or x^2\cos(1/x) are often handled by bounding rather than direct substitution. Since |\sin(1/x)|\le 1 for all x\ne 0, we have |x\sin(1/x)|\le |x|. The squeeze theorem then gives a limit of 0 as x\to 0.

The factor \sin(1/x) alone has no limit at 0 because it oscillates between -1 and 1. A product can still have a limit when one factor fails to converge, provided the other factor tends to zero. Standard limits such as \lim_{x\to 0}\dfrac{\sin x}{x}=1 and \lim_{x\to 0}\dfrac{1-\cos x}{x^2}=\dfrac12 should be memorised for quick evaluation.

Möbius transformation mapping

A Möbius transformation w=\dfrac{az+b}{cz+d} with ad-bc\ne 0 maps lines and circles in the z-plane onto lines and circles in the w-plane. To find the image of a set, either solve for z in terms of w and substitute, or test a few convenient points. For example, w=\dfrac{z-1}{z+1} maps the imaginary axis onto the unit circle, since z=jy gives |w|=\dfrac{|jy-1|}{|jy+1|}=1.