1. Transcendental functions
Chapter 3
Gialih Lin, Ph. D. Professor

2. 3.1 Concepts functions other than algebraic function
Transcendental functions
Trigonometric function三角函數
Exponential function指數函數
Logarithmic function對數函數
Hyperbolic function雙曲函數

3. 3.2 Trigonometric function
sinq =opposite/hypotenuse=BC/AC
cosq =adjacent/hypotenuse=AB/AC
tanq =opposite/adjacent =BC/AB
=sinq/cosq 斜邊 對邊 鄰邊

4. Pythagoras theory The right angle of ABC (900) AB2+BC2=AC2
cos2q+sin2q=1

5. Unit of angle Degree Radian (SI symbol rad)
The length of arc (s), radius (r)
s/2pr=q/3600
q = s x 3600/ r x 2p
1 rad = 3600/2p=57018’
q>0

6. Trigonometric functions for all angles
Quadrant 象限
Table 3.1 signs of the trigonometric functions
Negative angles
An angle is defined to have positive value for an anti-clockwise 逆時針rotation and negative for a clockwise 順時針rotation.

7. Further angles sin(-q)=sin(2p-q)=-sinq cos(-q+2pn)=cos(2p-q)=cosq
tan(-q)=tan(2p-q)=-tanq
sin(q±2pn)=sinq
cos(q±2pn)=cosq
tan(q±pn)=tanq
Curves are repeated p70 Figure 3.11

8. Periodic functions and harmonic waves
f(x±a)=f(x) a periodic function of x with period as; the value of the function is unchanged when x is replaced by x+a or x-a. The sine and cosine curves in Figure 3.11 are called harmonic waves, and the functions form the basis for the description of all forms of waves and other oscillatory motions.

9. Example 3.7 harmonic waves
f(x,t)=Asin2p[(x/l)-nt]
Wave length l 波長
Speed of propagation前進 v=ln
frequency 頻率or number of oscillations per unit time n
Period, the time of a complete oscillation t
t=1/ n 周期
Amplitude of the wave A 震幅

10. 15-1 Characteristics of Wave Motion
Wave characteristics:
Amplitude, A
Wavelength, λ
Frequency, f and period, T
Wave velocity,
Figure Characteristics of a single-frequency continuous wave moving through space.

11. Example 3.8 Newton’s second law of motion of thje simple harmonic oscillation (see also section 12.5 Figure 12.3 p350)
f=ma
-kx=m d2x/dt2
This is one of the simplest second-order differential equations (Chapter 12) and a solution is
x(t)=A cos wt
Angular frequency w= (k/m)1/ 角頻率
t=2p/w=1/ n 周期
n = (k/m)1/2 / 2p 頻率

12. 15-5 The Wave Equation Look at a segment of string under tension:
Newton’s second law gives:
Figure Deriving the wave equation from Newton’s second law: a segment of string under tension FT .

13. 15-5 The Wave Equation
Assuming small angles, and taking the limit Δx → 0, gives (after some manipulation):
This is the one-dimensional wave equation; it is a linear second-order partial differential equation in x and t. Its solutions are sinusoidal waves.

14. 3.3 Inverse trigonometric functions
If y=sinx then x=sin-1y
arcsin y=sin-1y
arccos y=cos-1y
arctan y=tan-1y

15. 3.4 Trigonometric relations
Sine rule
sinA/a=sinB/b=sinC/c
Cosine rule
a2=b2+c2-2bc cosA

16. Compound-angle identities
sin(x+y)=sinx cosy+cosx siny
sin(x-y)=sinx cosy-cosx siny
cos(x+y)=cosx cosy-sinx siny
cos(x-y)=cosx cosy+sinx siny
x=y
sin2x=2sinx cosy
cos2x=cos2x-sin2x=1-2sin2x=2cos2x-1

17. Example 3.14 sin((p/2)±q)=sin(p/2)cos q ±cos(p/2) sin q
sin(p/2)=1 and cos (p/2)=0
sin((p/2)±q)=cosq

18. Example 3.15 The harmonic wave
the harmonic wave travelling in the x-direction
f+=Asin2p[(x/l)-nt]
the harmonic wave travelling in the opposite direction
f-=Asin2p[(x/l)+nt]
As the waves overlap they interfere to give a new wave
y=af+ +b f- =a Asin2p[(x/l)-nt]+ bAsin2p[(x/l)+nt]
=(a+b)Asin(2px/l)cos(2pnt)-(a-b)Acos(2px/l)sin(2pnt)
For a=b=1
y=2Asin(2px/l)cos(2pnt)
This is called a standing wave. Its shape is given by the x-dependent factor 2Asin(2px/l), as shown in Figure 3.16.
There is constructive interference建設性干擾; the amplitude has double
節 nodes (zeros)

19. 15-4 Mathematical Representation of a Traveling Wave
Suppose the shape of a wave is given by:
Figure In time t, the wave moves a distance vt.

20. 15-4 Mathematical Representation of a Traveling Wave
After a time t, the wave crest has traveled a distance vt, so we write:
Or:
with ,

21. 15-6 The Principle of Superposition
Superposition: The displacement at any point is the vector sum of the displacements of all waves passing through that point at that instant.
Fourier’s theorem: Any complex periodic wave can be written as the sum of sinusoidal waves of different amplitudes, frequencies, and phases.
Figure The superposition principle for one-dimensional waves. Composite wave formed from three sinusoidal waves of different amplitudes and frequencies (f0, 2f0, 3f0) at a certain instant in time. The amplitude of the composite wave at each point in space, at any time, is the algebraic sum of the amplitudes of the component waves. Amplitudes are shown exaggerated; for the superposition principle to hold, they must be small compared to the wavelengths.

22. 3.5 Polar coordinates極座標 Point P in cartesian coordinate (x,y)
In polar coordinate (r,q)
x=rcosq
y=rsinq
r2=x2+y2
tanq=y/x
r=(x2+y2)1/2
q=tan-1(y/x) when x>0
q=tan-1(y/x)+p when x<0
Examples 3.17,3.18 p79

23. Ordinary geometry
Vector, r
r = a1i+a2j+a3k
Column vector

24. Figure 5-1
Figure 5-1 The relationship between spherical coordinates (r.θ.Φ) and Cartesian coordinates (x,y,z). Here θ is the angle with respect to the Cartesian z-axis, which ranges from 0 to π, and Φ is the azimuthal angle (the angle between the x-axis and the projection onto the x-y plane of the arrow from the origin to P), which ranges from 0 to 2π . Here, r is the distance of the electron from the origin, and ranges from 0 to ∞.
Fig. 5-1, p. 171

25. Figure 5-3
Figure 5-3 The differential volume element in spherical polar coordinates.
Fig. 5-3, p. 174

26. 3.6 Exponential function An exponential function f(x)=ax
The number a is the base and the variable x is the exponent.
10x
The exponential function
exp(x)=ex
The base is the Euler number
The Euler number e = 1+ 1/1! + 1/2! + 1/3!+ 1/4!+ … = =
Example 3.19

27. Example 3.20 Exponential growth and decay
Rate of growth system
dx/dt=±kx(t) (see Chapter 4)
+ for growth and – for decay; k is the rate constant
x(t)=x0e±kt
After time t
x=2t/tx0
k=ln2/t

28. Half-Life
Period for one-half of the original elements to undergo radioactive decay
Characteristic for each isotope
Fraction remaining =
n = number of half-lives

29. Practice Problems

30. The Euler number e and lifetime
Rate constant, k
Doubling time (halflife), T
Lifetime, t
[A]/[A]o =e±(t/t) = 2±(t/T) = e±kt
k = 1/t = ln2/T = 0.693/T
doubling time (halflife) T

31. Example 3.21 Atomic orbitals
The 1s orbital for an electron in the ground state of hydrogen atom is
y=e-r
Where r is the distance of the electron from nucleus. All the orbitals for the hydrogen atom have the form
y=f(x,y,z)e-r
Where (x,y,z) are the cartesian coordinates of electron relative to the nucleus at the origin, and a is a constant. The function f(x,y,z) is a polynomial in x,y, and z, and determines the shape of the orbital; for exasmple, f=z gives a pz orbital.

32. TABLE 5-2
Table 5-2, p. 175

33. Figure 5-4
Figure 5-4 Four representations of hydrogen sorbitals (a) A contour plot of the wave function amplitude for a hydrogen atom in its 1s, 2s, and 3s states. The contours identify points at which Ψ takes on ±0.05, ±0.1, ±0.3, 0.5, ±0.7, and ±0.9 of its maximum value. Contours with positive phase are shown in red; those with negative phase are shown in blue. Nodal contours, where the amplitude of the wave function is zero, are shown in black. They are connected to the nodes in the lower plots by the vertical green lines. (b) The radial wave functions plotted against distance from the nucleus, r. (c) The radial probability density, equal to the square of the radial wave function multiplied by r2. (d) The “size” f the orbitals, as represented by spheres whose radius is the distance at which the probability fall to 0.05 of its maximum value.
Fig. 5-4, p. 177

34. Figure 5-5
Figure 5-5 Two representations of hydrogen p orbitals. (a) The angular wave function for the pz orbital. The pz and py orbitals are the same, but are oriented along the x- and y-axis, respectively. (b) The square of the angular wave function for the pz orbital. Results for the px and py orbitals are the same, but are oriented along the x- and y-axis, respectively.
Fig. 5-5, p. 178

35. Figure 5-6
Figure 5-6 Radial wave functions Rnℓ for np orbitals and the corresponding radial probability densities r2R2nℓ.
Fig. 5-6, p. 179

36. Figure 5-7
Figure 5-7 Contour plot for the amplitude in the pz orbital for the hydrogen atom. This plot lies in the x-z plane. The z-axis (not shown) would be vertical in this figure, and the x-axis (not shown) would be horizontal. The lobe with positive phase is shown in red, and the lobe with negative phase in blue. The x-y nodal plane is shown as a dashed black line. Compare with figure 5.5a.
Fig. 5-7, p. 180

37. Figure 5-8
Figure 5-8 The shapes of the three 2p oritals, with phases and nodal planes indicated. The isosurfaces in (a), (b), and (c) identify points where the amplitude of each wave function is ±0.2 of its maximum amplitude. (a) 2pz orbital. (b) 2px orbital. (c) 2py orbital.
Fig. 5-8, p. 180

38. Example 3.33 The normal distribution
The normal or Gaussian distribution in statistics 統計by the probability density function
p(x)=(1/s(2p)1/2)exp[-(1/2)[(x-m)/s)]2]
The mean of the distribution m
The standard deviation of the distribution s
See Section 21.8

39. 3.7 The logarithmic function 對數
The logarithmic function (x=logay
) is the inverse function of the exponential
if y=ax then x=logay
The logarithm to base a of y
The ordinary logarithm to base 10
y=10x
x=log10y=logy=lgy
The natural logarithm (sometimes called the Napierian logarithm), to base e,
y=ex
x=logey=lny
log1=0
lnx+lny=ln(xy)
Lnx-lny=ln(x/y)
lnxn=nlnx

40. Example 3.28 pH as a measure of hydrogen-ion concentration
pH=-log[H+]
[H+]=10-pH mol dm-3
M=mol/L= mol dm-3

41. The logarithm as ascale of measure
Richter scale for the strength of earthquake 芮氏地震強度
The bel scale of loudness 音貝
The scale of star magnitude

42. Conversion of factors logbx=logba logax
lgx=log10x= log10elogex=0.434 lnx
lnx= loge10log10x=2.3 logx

43. 3.8 Values of exponential and logarithmic functions
Table 3.8 p86-97
Example 3.29 p87

44. 3.9 Hyperbolic functions 雙曲線函數
Hyperbolic functions have their origin in geometry in the description of the properties of hyperbola.
y=1/x
y=ax/(b+x)

45. Leonor Michaelis & Maud Menton

46. Fig 5.3 Plots of initial velocity (vo) versus [S]
(a) Each vo vs [S] point is from one kinetic run
(b) Michaelis constant (Km) equals the concentration of substrate needed for 1/2 maximum velocity

47. The Michaelis-Menten equation
Equation describes vo versus [S] plots
Km is the Michaelis constant
Vmax[S]
vo =
Km + [S]

48. Hyperbolic functions Hyperbolic cosine and sine coshx=(ex+e-x)/2
sinhx=(ex-e-x)/2
Red cosh and shine
cosh2x-sinh2x=1
sinh(x±y)=sinhx coshy±coshx sinhy
cosh(x±y)=coshx coshy±sinhx sinhy

49. Inverse hyperbolic functions
cosh-1x=ln[x± (x2-1)1/2] (x≧1)
sinh-1x=ln[x+ (x2+1)1/2]
tanh-1x= ½ ln[(1+x)/(1-x)] (∣x ∣<1)