Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.8 APPROXIMATIONS TO INTEGRALS to the l.s.d., i.e. on the imaginaryt-axis, provides some reassurance. Whetherμis positive or n ...

APPLICATIONS OF COMPLEX VARIABLES Finally, putting the various values into the formula yields F(x)∼+ ( 2 π A ) 1 / 2 g(i)exp[f(i ...

25.8 APPROXIMATIONS TO INTEGRALS which we already know has the value √ πwhenzis real. This choice of demon- stration model is no ...

APPLICATIONS OF COMPLEX VARIABLES from which it follows thatC(∞)=S(∞)=^12. Clearly,C(−∞)=S(−∞)=−^12. We are now in a position to ...

25.8 APPROXIMATIONS TO INTEGRALS (a) (b) (c) β π/ 4 √ π Figure 25.13 Amplitude–phase diagrams for the integral ∫∞ −∞exp(−z (^2) ...

APPLICATIONS OF COMPLEX VARIABLES stationary, the magnitude of any factor,g(z), multiplying the exponential function, exp[f(z)]∼ ...

25.8 APPROXIMATIONS TO INTEGRALS are self-cancelling, as discussed previously. However, the ends of the contourmust be in region ...

APPLICATIONS OF COMPLEX VARIABLES (a) (b) (c) (d) v v v Figure 25.14 Amplitude–phase diagrams for stationary phase integration. ...

25.8 APPROXIMATIONS TO INTEGRALS
In the worked example in subsection 25.8.2 the function F(x)= 1 π ∫∞ 0 cos(^13 s^3 +xs)ds (∗) ...

APPLICATIONS OF COMPLEX VARIABLES V 0 eiωt ̃ IR L L C C A B DE R Figure 25.15 The inductor–capacitor–resistor network for exerci ...

25.9 EXERCISES imaginaryz-axes, find the strengths of the field (a) at a point one metre directly above the fence, (b) at ground ...

APPLICATIONS OF COMPLEX VARIABLES Then, by the principle of the argument, the number of zeros insideCis given by the integer (2π ...

25.9 EXERCISES (b) CalculateF(s) on either side of the branch cut, evaluate the integral and hence determinef(t). (c) Confirm th ...

APPLICATIONS OF COMPLEX VARIABLES 25.19 The functionh(z)ofthecomplexvariablezis defined by the integral h(z)= ∫i∞ −i∞ exp(t^2 − ...

25.10 Hints and answers t=−iand then approaches the origin in the fourth quadrant in a curve that is ultimately antiparallel to ...

APPLICATIONS OF COMPLEX VARIABLES 25.17 Use the binomial theorem to expand, in inverse powers ofz, both the square root in the e ...

26 Tensors It may seem obvious that the quantitative description of physical processes cannot depend on the coordinate system in ...

TENSORS 26.1 Some notation Before proceeding further, we introduce thesummation conventionfor subscripts, since its use looms la ...

26.2 Change of basis In the second of these the dummy index shared by both terms on the left-hand side (namelyj) has been replac ...

TENSORS Scalars behave differently under transformations, however, since they remain unchanged. For example, the value of the sc ...