1000 Solved Problems in Modern Physics

1.2 Problems 21

Extract the square root of both members, separate the variables, and integrate
again, introducing the second arbitrary constantC 2.

Complex variables

Complex numberz=r(cosθ+isinθ), wherei=

√

− 1

zn=cosnθ+isinnθ

Analytic functions

A functionfof the complex variablezis analytic at a pointzoif its derivativef′(z)
exists not only atzobut at every pointzin some neighborhood ofzo. As an example
iff(z)=^1 zthenf′(z)=−z^12 (z
=0). Thusfis analytic at every point except the
pointz=0, where it is not continuous, so thatf′(0) cannot exist. The pointz= 0
is called a singular point.

Contour

A contour is a continuous chain of finite number of smooth arcs. If the contour
is closed and does not intersect itself, it is called a closed contour. Boundaries of
triangles and rectangles are examples. Any closed contour separates the plane into
two domains each of which have the points ofCas their only boundary points. One
of these domains is called the interior ofC, is bounded; the other, the exterior ofC,
is unbounded.
Contour integral is similar to the line integral except that here one deals with the
complex plane.

The Cauchy integral formula

Letfbe analytic everywhere within and on a closed contourC.Ifzois any point
interior toC, then

f(zo)=

1

2 πi

∫

C

f(z)dz

z−zo

where the integral is taken in the positive sense aroundC.

1.2 Problems..................................................

1.2.1 VectorCalculus....................................

1.1 Ifφ=^1 r, wherer=(x^2 +y^2 +z^2 )^1 /^2 , show that∇φ=rr 3.

1.2 Find a unit vector normal to the surfacexy^2 +xz=1 at the point (− 1 , 1 ,1).

1.3 Show that the divergence of the Coulomb or gravitational force is zero.

1.4 IfAandBare irrotational, prove thatA×Bis Solenoidal that is div (A×B)=

0