Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 7 ppt

Appendix BNotation C class of continuous functions Cn class of n-times continuously differentiable functions C set of complex numbersδx Dirac delta function F [·] Fourier transform Fc[·]

0

)2− z000 = 0Multiply by z00 and integrate

The linearized equation is

ut+ 30u2u1+ 20u1u2+ 10uu3 + u5 = 0

We make the substitution

u(x, t) = z(X), X = x − U t

−U z0 + 30z2z0+ 20z0z00+ 10zz000+ z(5) = 0Note that (zz00)0 = z0z00+ zz000

−U z0+ 30z2z0+ 10z0z00+ 10(zz00)0+ z(5) = 0

−U z + 10z3+ 5(z0)2+ 10zz00+ z(4) = 0Multiply by z0 and integrate

(20 + 40a + 15a2)z4+ (40b + 20ab)z3+ (4b2+ 4U )z2 = 0This equation is satisfied by b2 = U , a = −2 Thus we have

The linearized equation is

ut+ u5 = 0.Substituting u = e−αx+βt into this equation yields

β − α5 = 0

We set

α = U1/4.The solution for u(x, t) becomes

Part VIII Appendices

Appendix B

Notation

C class of continuous functions

Cn class of n-times continuously differentiable functions

C set of complex numbersδ(x) Dirac delta function

F [·] Fourier transform

Fc[·] Fourier cosine transform

Fs[·] Fourier sine transform

γ Euler’s constant, γ =R∞

0 e−xLog x dxΓ(ν) Gamma function

H(x) Heaviside function

Hν(1)(x) Hankel function of the first kind and order ν

Hν(2)(x) Hankel function of the second kind and order ν

−1

Jν(x) Bessel function of the first kind and order ν

Kν(x) Modified Bessel function of the first kind and order νL[·] Laplace transform

N set of natural numbers, (positive integers)

Nν(x) Modified Bessel function of the second kind and order ν

R set of real numbers

R+ set of positive real numbers

R− set of negative real numbers

o(z) terms smaller than z

O(z) terms no bigger than z

Appendix C

Formulas from Complex Variables

Analytic Functions A function f (z) is analytic in a domain if the derivative f0(z) exists in that domain

If f (z) = u(x, y) + ıv(x, y) is defined in some neighborhood of z0 = x0+ ıy0 and the partial derivatives of u and vare continuous and satisfy the Cauchy-Riemann equations

ux = vy, uy = −vx,then f0(z0) exists

Residues If f (z) has the Laurent expansion

Residue Theorem Let C be a positively oriented, simple, closed contour If f (z) is analytic in and on C exceptfor isolated singularities at z1, z2, , zN inside C then

, 0



Residues of a pole of order n If f (z) has a pole of order n at z = z0 then

Res(f (z), z0) = lim

z→z 0

1(n − 1)!

dn−1

dzn−1 [(z − z0)nf (z)]



Jordan’s Lemma

Z π 0

e−R sin θ dθ < π

R.Let a be a positive constant If f (z) vanishes as |z| → ∞ then the integral

Z

C

f (z) eıaz dzalong the semi-circle of radius R in the upper half plane vanishes as R → ∞

Taylor Series Let f (z) be a function that is analytic and single valued in the disk |z − z0| < R

n

The series converges for |z − z0| < R

Laurent Series Let f (z) be a function that is analytic and single valued in the annulus r < |z − z0| < R In thisannulus f (z) has the convergent series,

cn= 1ı2π

I

f (z)(z − z0)n+1dzand the path of integration is any simple, closed, positive contour around z0 and lying in the annulus The path ofintegration is shown in Figure C.1

dxarccosh x =

1

√

x2− 1, x > 1, arccosh x > 0d

dxarctanh x =

1

1 − x2, x2 < 1

∂f (ξ, x)

∂x dξ + f (h, x)h

0− f (g, x)g0

f (x) dx = log f (x)Z

f0(x)2pf(x)dx =

1

xdx = log xZ

eax dx = e

ax

a

x2− a2 dx =

(

1 2aloga−xa+x for x2 < a2

1 2alogx−ax+a for x2 > a2

cos(ax) dx = 1

asin(ax)

tan(ax) dx = −1

alog cos(ax)Z

csc(ax) dx = 1

alog tan

ax2Z



Z

cot(ax) dx = 1

alog sin(ax)Z

sinh(ax) dx = 1

acosh(ax)Z

cosh(ax) dx = 1

asinh(ax)Z

tanh(ax) dx = 1

alog cosh(ax)Z

csch(ax) dx = 1

alog tanh

ax2Z



Z

coth(ax) dx = 1

alog sinh(ax)

x sin ax dx = 1

a2 sin ax − x

acos axZ

x2sin ax dx = 2x

a2 sin ax −a

2x2− 2

a3 cos axZ

x cos ax dx = 1

a2 cos ax + x

asin axZ

x2cos ax dx = 2x cos ax

a2 + a

2x2− 2

a3 sin ax

Appendix F

Definite Integrals

Integrals from −∞ to ∞ Let f (z) be analytic except for isolated singularities, none of which lie on the realaxis Let a1, , am be the singularities of f (z) in the upper half plane; and CR be the semi-circle from R to −R inthe upper half plane If

Integrals from 0 to ∞ Let f (z) be analytic except for isolated singularities, none of which lie on the positive realaxis, [0, ∞) Let z1, , zn be the singularities of f (z) If f (z)  zα as z → 0 for some α > −1 and f (z)  zβ as

z → ∞ for some β < −1 then

Z ∞ 0

ν+2n

|z| < ∞

Appendix I

Continuous Transforms

Let f (t) be piecewise continuous and of exponential order α Unless otherwise noted, the transform is defined for s > 0

To reduce clutter, it is understood that the Heaviside function H(t) multiplies the original function in the following twotables

f (t)

Z ∞ 0

e−stf (t) dt

1ı2π

Z c+ı∞

c−ı∞

etsf (s) dsˆ f (s)ˆ

af (t) + bg(t) a ˆf (s) + bˆg(s)d

f (t) dt

I.2 Table of Laplace Transforms

f (t)

Z ∞ 0

e−stf (t) dt

1ı2π

I.3 Table of Fourier Transforms

12π