In a similar way each of the other inverse ftmctions is defined We find cos—ls--i log (s -1-1/10."..-1)9 S tan--1/4-= — Log 7---• 2 From the foregoing it appears that in the domain of complex quantities the trigonometric functions can be expressed in simple form in terms of the exponential function and con versely; while the inverse trigonometric func tions express themselves in terms of the loga rithm. Thus the formulas of integration: , taxi (x4D2 ab>0; dx ab •fa + 1—b ab< 0, — log t --ab • \ / _b apparently in no wise akin, are seen to be equiv alent in the complex domain.
6. Laplace's Equation.—(a) Flow of Heat or Electncity. Let the points of the boundary of a homogeneous (or more properly isotropic) conducting substance be maintained at given temperatures (for example, a steam-pipe, the interior being kept at 100° arid the exterior at 0°). Then a flow of heat ensues within the conductor and the temperature at any given point approaches a limiting value, u, as time goes on. In fact, if each point were brought to this limiting temperature, it would continue there so long as the boundary conditions are maintained and we should have a steady flow. The corresponding temperattire u amis. fies Laplace's Equation in three dimensions: a1/4 „ =.- -t- --- - ax2 ay, 80, • If in particular the solid be bounded by cylin drical surfaces whose elements are parallel to the z-axis, and if the given temperature be con stant along each element, then u will evidently be constant along any line parallel to the z axis, and we shall have in substance a two dimensional flow, for which du We may conceive the solid itself in which •the flow takes place as two-dimensional if we think of a thin section as cut out of the above solid by two planes parallel to the (x, y)-plane and very near together. The flat faces of this slab are then to be thought of as coated with some adiabatic substance, so that no heat can enter or leave the slab across these faces.
The electrical problem is mathematically identical with the heat problem, u being in terpreted as potential instead of temperature. As conductor consider for example a piece of tin foil whose edge is connected with a thick piece of copper. Let one pole of a battery be at tached to the copper and the other to the tin foil at an interior point. We thus have a point source.
The curves u=const. are called isothermals
or equipotential curves. The orthogonal trajec tories of these curves v..=const. are the lines of flow.
(b) Flow of an Incompressible Fluid. Con sider the flow of an incompressible fluid con strained to move between two parallel planes, the particles which lie in any line perpendicular to the planes at any moment always remaining in that line. If there be a function u whose partial derivatives with regard to x and v give the components of the velocity of each particle of the fluid along the axes, then u is called the velocity potential. If u does not depend on t, we have a steady flow with velocity potential, and u here satisfies Laplace's Equation.
Conversely, let u be any harmonic function, i.e., a solution of Laplace's Equation, in two dimensions. Then this function defines a steady flow of heat, electricity, or an incompressible fluid with velocity potential.
Let us turn now to analytic functions of a complex variable: 4 -Fv1=1 ( .
Differentiating the first of the equations (1) with regard to x, the second with regard to y, and adding, we see that u satisfies Laplace s Equation. Hence Theorem: Every analytic function of a complex variable defines a steady flow of heat, electricity, or an incompressible fluid.
The converse is also true. Let us first recall the condition that P dx + Q dy be an exact differential, namely, aP ay ax If this condition is fulfilled throughout a finite region S, then taken over the complete boundary of S will vanish. Furthermore, if S has but a single boundary, rx, dx Q dy (a, 8) will be a function of the upper limit (x, y) alone, not of the path of intetion; cf. Gour sat, d'analyse,) VoL I, § 1 It can now be shown that, given a steady flow of heat or electricity, a possible flow results if we inter change the equipotential lines and lines of flow.
Returning now to the main question, let u be any solution of Laplace's Equation, and form the function (x. )) au v - dx + dy+ C.
(a, b) ax Then the value of the integral will be inde pendent of the path of integration,— at least if we confine ourselves to a finite region S with a single boundary. The derivatives of this func tion have the values av au av au , ax —I ax ay ay and thus satisfy (1). Herice u and v are con jugate functions, and u+vi is an analytic func tion of x+yi.