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PreviewPreview:: Preliminary and Engineering Problems
Binomial theoremBinomial theorem
EulerEulers formulas formula
DifferentiationDifferentiation
IntegrationIntegration
Complex variablesComplex variables
Tensor and vector calculusTensor and vector calculus
Hyperbolic functionsHyperbolic functions
Simplest Differential EquationsSimplest Differential Equations
Professor K.T. Chau
P-1
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2 3
2 3
2 3
( 1) ( 1)( 2)lim (1 ) lim[1 ( ) ( ) ( ) ...]1! 2! 3!
(1 1/ ) (1 1/ )(1 2 / )lim[1 ...]
1! 2! 3!
1 ...1! 2! 3!
x n
n n
n
x n x n n x n n n xe n n n n
x n x n n x
x x x
= + = + + + +
= + + + +
= + + + +
1 2 2 1( 1)( )2
n n n n n nn nx h x nx h x h nxh h + = + + + + +
Binomial theorem
Proof of power series ofex
Review on fundamentals (something you MUST know)
0
1
2 2 2
3 3 2 2 3
4 4 3 2 2 3 4
( ) 1
( )
( ) 2
( ) 3 3
( ) 4 6 4
...
a b
a b a b
a b a ab b
a b a a b ab b
a b a a b a b ab b
+ =
+ = +
+ = + +
+ = + + +
+ = + + + +
Pascals triangle (1654)
Apianus (1527)
(1261)
Jia XianPascal
Yang Hui
P-2
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1 2 2 1
1 1 2 2 2 1 1
( 1)( )
2
...
n n n n n n
n n n r n r r n n nn n n n
n nx h x nx h x h nxh h
x C x h C x h C x h C xh h
+ = + + + + +
= + + + + + + +
( 1)...( 1) !
! !( )!r
n
n n n r nC
r r n r
+= =
Example P-1
1
1 1
( 1)! ( 1)!
( 1)!( )! !( 1 )!
( 1)![ ( )]
( 1)!( )!
!!( )!
r r
n n
rn
n n
C C r n r r n r
nr n r
r n r
n Cr n r
+ = +
= +
= =
Basic identity
Related to probability
Problem P-1 Find S?
1 2 11 ... 1r nn n n nS C C C C = + + + + + + +
P-3
Hua Luogeng
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Review on compound interest and the birth ofe
Problem P-2 FindS1?1 2 1 1
1 1 ... ( 1) ( 1)n n n
n n nS C C C = + + + +
Growth of money in bank
(1 )S P r= + r = annual interest ratePr= interest at the end of 1 year
After year, take out money and redeposit (a bigger P)
2[ (1 )](1 ) (1 )2 2 2r r rS P P= + + = +
After 1/n year, take out money and redeposit (a bigger P)
[ (1 )...](1 ) (1 )nr r r
S P Pn n n= + + = +
The best you can earn in compound interest (in 1 year)
Compound interest
lim (1 )n rn
rS P Pe
n= + =
1lim (1 ) 2.71828...nn
en= + =
P-4
Euler
Eulers number
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Gerolamo Cardano
(1501-1576)
Italian mathematician Gerolamo Cardano is the firstknown to have introduced complex numbers in 1545.He called them "fictitious", during his attempts to find
solutions to cubic equations in the 16th century
History of Complex Numbers
3 2 0ax bx cx d + + + =
1i =
General formula of roots for cubic equation
P-5
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Review on Eulers formula
(cos sin )ix iy re r i + = = +
Demoivres Formula
Example P-2 Special case ofr=1 and =
1ie = 1 0ie + = Eulers formula
Problem P-3 Find ?ii =(Answer: Infinite answers and the smallest one is 0.207879576)
Problem P-4 Find ?i i =(Answer: Infinite answers and the smallest one is 4.810477381)
1i =
(cos sin ) cos sinni n i n + = +
P-6
LHopitals Rule (actually by Johann Bernoulli)
( ), ( )f x g x
( ), ( ) 0f x g x
( ) ( )
lim lim( ) ( )x a x a
f x f x
g x g x
=
x a( ), ( )f x g x
are differentiable
0,
0
{ }( ) ( 1) ( 2) (1) ( 3) (2) ( 1) ( 1) ( )
... ( 1) ( 1)k k k k k k k k d
VU U V U V U V UV UV dx
= + + +
LHopitalBernoulli
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Swiss mathematician and scientist
Euler (1707-1783)
Saturday afternoon lessons from
Johann Bernoulli
Johann Bernoulli
(1667 1748)
Basel University(1459)
the oldest universityin Switzerland
Euler cannot get a job
at Basel University
Daniel Bernoulli
Father-son
friends
Teacher & inspirer
Nicolaus II Bernoulli
Catherine I
Russia
1727
Frederick II
Prussia
1741
P-7
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Swiss franc
P-8
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Letters of Euler on different Subjects in Natural Philosophy
Addressed to a German Princess
http://www.math.dartmouth.edu/~euler/tour/tour_00.html
886 publications of Euler available here
German Princess
P-9
Seven Bridges of
KnigsbergGraph theory
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Review on differentiation
( ) lnu ud dua a adx dx=1log lnad duudx u a dx=1lnd duudx u dx=u ud du
e edx dx=
Derivatives of Exponential and Logarithmic Functions
( ) 1n nd
nx
dx
=( ) 0d
cdx
=( )( ) ( )
0
limh
f x h f xdff x
dy h
+ = =
( )d dv du
uv u vdx dx dx
= +
DefinitionPower rule
( )d du
cu cdx dx
=
2
du dvv u
d u dx dx
dx v v
=
Quotient ruleProduct rule
( )sin cosd du
u udx dx
= ( )cos sind du
u udx dx
=
Circular functions (Trigonometric Functions )
( ) 2tan secd du
u udx dx
=
Chain Rule: dy dy dudx du dx
=
(sinh ) coshd du
u udx dx
= (cosh ) sinhd du
u udx dx
= 2(tanh ) sechd du
u udx dx
=
Hyperbolic functions
( ), ( )y y u u u x= =
1
2
1sin
1
d duu
dx dxu
=
1
2
1tan
1
d duu
dx u dx
=+
1
2
1cos
1
d duu
dx dxu
=
1
2
1(sinh )
1
d duu
dx dxu
=+
1
2
1(cosh )
1
d duu
dx dxu
=
1
2
1(tanh )
1
d duu
dx dxu
=
( ) 1n nd du
u nu
dx dx
=
( )d du dv
u vdx dx dx
+ = +
Sum rule Constant multiple
P-10
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Leibniz's rule of differentiation under integral sign
( ) ( )
( ) ( )
( , ) ( ) ( )( , ) [ , ( )] [ , ( )]
h x h x
g x g x
d df x dh x dg xf x d d f x h x f x g x
dx dx dx dx
= +
General Leibniz rule
0
( )n k n k n
k
nn k n k k
d d f d g fg C
dx dx dx
=
=
Example P-3 9 9 8 2 79 9 8 2 7
sin sin 9 8 sin( sin ) 9 ( ) ( ) ...2!
cos 9sin
d d x d d x d d xx x x x xdx dx dx dx dx dx
x x x
= + + += +
0
Partial differentiation0
0
( , ) ( , )lim ( )
( , ) ( , )lim ( )
y xx
x yy
f f x x y f x y ff
x x x
f f x y y f x y f
y y y
+ = = =
+ = = =
( , ) ( , )
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )[ ] [ ]
0 0
and
as and , the total differential i
x x x y y y f f f
f f x x y y f x y
f x x y y f x y y f x y y f x y
f x x y y f x y y f x y y f x yx y
x y
x y df
+ + +
= + +
= + + + + + + + + +
= +
s
f fdf dx dyx y
= +
Total differential
f f
df dx dyy
= +
Proof
1 2
1 2
... nn
f f fdf dx dx dx
x x x
= + + +
P-11
Leibniz
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( , ) ( ) ( )and ,f f x y x x u y y u
f df f dx f dydf dx dy
x y du x du y du
= = =
= + = +
Chain rule
Problem P-5 Show that
Polar coordinates and, Cartesian coordinates x and y, x=cos ,y=sin, transform into one in and
2 2 2
1 2 1 2 1 2 2 1 1 2
( ) 2 2v u v u v u
uv u vx x x x x x x x x x
= + + +
2 2 2
2 2 1/2
21
2 2 2 2
2
2
cos sin( )
/ sin sin costan
1 ( / )sin cos
cos sin
( )
, ,
), ,
,
and
xx y
x x y y
y x y (y / x
x y x x y y
x x x y
f f
x x x
= + = = =
+
= = = = = =
+ +
= + = = +
=
2 2 2 22
2 2 2 2 2 2
1 1( ) ( , )
2 f f f f f ff x y
y y y x y
= = + = + +
2 2
2 2
f
x y
+
( , ) ( , )f x y f
Example P-4
P-12
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( ) ,!
)()(
10
0)(
=
=n
nn
xxn
xfxfTaylor series expansion
"++=!6!4!2
1)cos(642 xxx
x "++=!7!5!3
)sin(753 xxx
xx"++++=!3!2
132 xx
xex
MaClaurin series ( )( )
1
(0)( )!
nn
n
ff x xn
=
=
(4) (4)
( ) sin (0) 0
'( ) cos '(0) 1
''( ) sin ''(0) 0
'''( ) cos '''(0) 1
( ) sin (0) 0
f x x f
f x x f
f x x f
f x x f
f x x f
= =
= =
= =
= = = =
2 3
3 5 7
2 1 3 5 7
0
'(0) ''(0) '''(0)(0)
1! 2! 3!
3! 5! 7!
sin ( 1)(2 1)! 3! 5! 7!
n
n
n
f f ff x x x
x x xx
x x x xx xn
+
=
+ + + +
= + +
= = + + +
Example P-5 ( ) sinf x x=
Examples
Expand
Example P-6
2 2 3 3 4 4 5 5
2 4 3 5
12! 3! 4! 5!
1 ( )2! 4! 3! 5!
ix i x i x i x i xe ix
x x x xi x
= + + + + + +
= + + + + +
"
" "
Second approach
Euler formula
"++=!6!4!2
1)cos(642 xxx
x
"++=!7!5!3
)sin(753 xxx
xx
( ) xf x e=
( ) ( ) ( ) ( ) ... xf x f x f x f x e = = = = =
2 3
2 3
'(0) ''(0) '''(0)(0)
1! 2! 3!
12! 3!
x f f fe f x x x
x xx
= + + + +
= + + + +"(0) (0) (0) (0) ... 1f f f f = = = = =
cos sinixe x i x= +
All 3 formulas are derived
P-13
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Review on integration
Multiple integral
2b
aV y dx= = 2
d
cV x dy
u dv uv v du= Integration by parts
Definition Summing area under a function
Finding volume
P-14
( ) ( 1) ( 2) ( 3) ( )...( 1)n n n n n ng dx f g f g f g fg dx = +
Generalized integration by parts
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General rule of integration
1( ) ( )ax dx f u du
a
=
( ){ ( )} ( )
( )
dx F uF f x dx F u du du
du f x= =
( )u f x=
1
1
n
n uu dxn
+
=+ 1n
1lndu u
u=
u ue du e=
lnln
ln ln
u a uu u a e a
a du e du a a= = = 0, 1a a>
Transformation rule
1( ) ( )F ax b dx F u du
a+ = u ax b= +
2( ) ( )F ax b dx uF u dua
+ = u ax b= +1( ) ( )nn
nF ax b dx u F u du
a
+ = nu ax b= +
2 2( ) ( cos )cosF a x dx a F a u udu = sinx a u=2 2 2( ) ( sec )secF a x dx a F a u udu+ = tanx a u=2 2( ) ( tan )sec tanF x a dx a F a u u udu =
secx a u=
P-15
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1 ( )( )ax
F ue dx du
a u=
axx e=
(ln ) ( ) uF x dx F u e du= lnu x=1(sin ) ( )cosx
F dx a F u udua
= 1sin
xu
a
=
2
2 2 2
2 1(sin ,cos ) 2 ( , )
1 1 1
u u duF x x dx F
u u u
=+ + + tan 2
xu =
Definite integrals
( ) ( ) ( ),
b
a f x dx b a f c= a c b<
( )1
! ( )( )
2 ( )
n
nC
n f zf z dz
i z a +=
>Remarkable results: Value of
f(z) and its higher derivatives
only depends on boundary
values on C
Singular points
( )f a is not analytic then a = isolated singular point
Poles
( )( )( )n
zf zz a
= z = a is a pole of ordernz = a is a simple pole ifn =1
( ) 0a
Example P- 82
( )( 3) ( 1)
zf z
z z=
+has two singularities, a pole of order 2 at z=2
and a simple pole atz= 1.
(orf(z) has singularity at a)
P-18
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21 10 1 21
( ) ... ( ) ( ) ... ( )
( )( ) ( )
kn nkn n
k
a a af z a a z a a z a a z a
z az a z a
+
=
= + + + + + + + =
Principal part Analytic part
This the principal part has infinite terms, it is an essential singularity
Example P-91/
2
1 11 ...
2!
zez z
= + + +
For simple pole
1
1 1
1lim {( ) ( )}
( 1)!
nn
nz a
da z a f z
n dz
=
a1 = residue
1 lim( ) ( )z a
a z a f z =
has is an essential singularity at z = 0
Residue Theorem
1 1 1( ) 2 ( ...)
Cf z dz i a b c
= + + +
>
C
ab
c
Laurents series
Residue
P-19
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Branch point & Branch cut nw z=
z planew plane
2
n
Branch cut
0-x is a branch cut (restrict to ensure single-valuedness)
1/ /n i nw e =
1/n
2
/ n
iz e =
n different points in w-plane corresponds
to the same point inz-plane
0 1 1{ } ,{ } ,...,{ }n n n
nz z z
nbranches of single-valued functions
Removable singularity
3 5 2 4
2 4
sin sin( ) sin 1( ...) 1 ...
3! 5! 3! 5!
( ) ( )1 ...3! 5!
z u u u u u uu
z u u u
z z
+= = = + = + +
= + + +
z u =Example P-10
(no singularity)
P-20
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Example P-11 Show that1
0, 0 1
1 sin
px
dx px p
= <
Rr
y
xBE
GHJ
1
C
D
1 1 2 1 12 0( 1)
20 2
(R ) ( ) ( )2
1 1 R 1 1
p i p i i p i p iR rp i
i i ir R
x e iR d xe dx re ir ddx ie
x e xe re
+ + + =+ + + +
idz iR d
=
idz ir d
=
, 1 ... 0
BDEFG
R p < 0, 0 ... 0HJA
r p >
1 2 ( 1) 10( 1)
02
1 1
p i p pp ix e x dx
dx iex x
+ =
+ + 1
2 ( 1) ( 1)
0[1 ] 2
1
pi p p ixe dx ie
x
=+
1 ( 1)
2 ( 1)0
2 2
1 1
sin
p p i
i p p i p i
x ie idx
x e e e
p
= =
+
=
QED
P-21
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First rank or order(e.g. displacement, velocity)
zeroth rank or order(e.g. temperature, pressure)
Second rank or order(e.g. stress, strain)
Third rank or order(e.g. permutation tensor )
1850 1919
Woldemar Voigt
Elementary Tensor Analysis
Fourth rank or order(e.g. elastic tensor )
Scalar (independent of direction)
3
1 1 2 2 3 3
1
u e e e e ei i i ii
u u u u u
== + + = =
1st order (vector) (e.g. displacement)
Einstein notation
(drop summation)
3 3
1 1
e e e eij i j ij i ji j
= =
= =
2nd order (e.g. stress, strain)
2nd order tensor (e.g. stress)
There are two direction senses
1x
2x
3x
3e1e
2e
ij
Plane i Directionj
Plane 1
Plane 3
3eNormal vector on plane 3 is
Plane 2
3 3 3 3
1 1 1 1
e e e e e e e eijkl i j k l ijkl i j k l
i j k l
C C
= = = =
= =
C
4th order tensor (stiffness tensor)
We need 9 components to fullydescribe stress at a point!!
1,2,3; 1,2,3i j= =
3 3 9 =
3 components to fully describe a vector
3 3 3 3 81 =
P-22
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Heaviside O.
(1850-1925)Riemann, G.F.B.
Vector analysis
Gibbs, J.W.
(1839-1903)
Riemann Hypothesis
The Clay Mathematics Institute (Cambridge, Massachusetts)
$1 million award
This formula says that the zeros of the Riemann zeta function controlthe oscillations of primes around their "expected" positions.
u(u1,u2,u3)
e2
e3
e1
3
1 1 2 2 3 31
u e e e e ei i i ii
u u u u u== + + = =
(1826 1866)Age =39
odd
even
13
2
1( )( )( )2ijke i j j k k i=
This is not a tensor equation!
123 231 312 1e e e= = =
132 213 321 1e e e= = =
Permutation tensor
Even permutation
Odd permutation
133 221 131 0e e e= = =
Story
1st PhD in USA in
engineering in 1863
P-23
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A Brief Review of Vector Analysis
0|cos ( ) | u v | = | u | | vdot product
i j ij =e eKronecker delta
Leopold Kronecker
(1823 1891)
German
3
1 1 2 2 3 3
1
k k k k k
u v u v u v u v u v
=
= + + = =u v
1 1 2 2 3 3 1 1 2 2 3 3= e + e + e , = e + e + eu u u v v vu v
| || | sin (0 / 2) = = w u v u v
cross product
i i ijk j k iw = e u v=w e e
Repeated indices in component
form means summation
vue=w kjijki
Polyadic form
component form
kis a dummy index (repeated)
jis a dummy index (repeated)
iis a free index (not repeated)
Rule in tensor notationRule in tensor notation
i imn m nw e u v=
same
Balance in free index on both size of =. 0,
1,
ij i j
i j
=
= =
P-24
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ikjijkii vue=w ee
u v = (v u)
= + + u (v w) u v u w
u u = 0
=
=
=
1 2 3
2 3 1
3 1 2
e e e
e e e
e e e
1e
2e
3e
( )k k k = u v = u v u v
2 3 2 3 1 3 1 1 3 2 1 2 2 1 3( ) ( ) ( )e e e= u v u v u v u v u v u v = + + w u vProblem P-6 Show that these are the same
1 2 3
1 2 3
1 2 3
2 3 3 2 1 3 1 1 3 2 1 2 2 1 3( ) ( ) ( )
e e e
u v
e e e
u u uv v v
u v u v u v u v u v u v
=
= + +
1 2 3det( )ij i j k ijke A A A=
ijk irs jr ks js kr e e = e-
identity
determinant
P-25
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Derivatives of tensors
=ix
i
e = (x,t)
ij ijii, j ij, k ij, j
j k j
v = , = , =vx x x
comma-subscript convention
Vector differential operator
,grad i iix
= = =
ie e
,div i iv= =v v
,curl ijk k jv e v= = iv e
2,ii = =
( )fg f g g f= + 2 2 2( ) 2( ) ( )fg f g f g g f = + +
( ) ( )f f f = +v v v 2( )f g f g f g = +
Identities exist for the differential operator
( ) 0f =( ) 0 =v
( ) ( = a b a) b a b ( )f f f = + v v v
2( ) ( = v v) v 2 2 )( = a a
2 2( ) ( ) = 2 2( ) 2 = a r a + r a 2 2( ) 2 = + r r
Gradient
Curl
Divergence
Laplacian
Vector calculus P-26
Useful formulas
1 2 31 2 3
=x x x
+ +
e e e
P 27
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( ) ( )k i j jk
= =e e e eij ij,i x
, ,( ) ( ) ( )i j k i j k k = =e e e e e e e ejk jk i llij jk ii
ex
=
n T T
VS
dS dV =
The Divergence Theorem
( )C S
d = dS T s T nStokes Theorem.C
S
n
dSds
V
S
n
dS
Some Formulae in Cylindrical Coordinate
1 2 3 cos sin1 2 3 1 2 3r + + + +e e e e e ex x x r r z= = 1
r=e
h x
x
=h
r
er
er
ez
r
11 1
cos sin sin cos
13
re
r
z
= = + , = = +
rh h
= =zh
r 2 1 2
z
r re e e e e e
e
rr
ee= , =e e
P-27
S f l i li d i l di t P 28
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( )( )
1 1(
1( )
r z r z z
r r z z z z z r
r z r r
= + + + e +e e e e er z
= + + + + +e e e e e e e e e e e e
+ +e e e e e e
r
r z z r r
z r
u u ur
u uu u u u+ )u
r r z r z z
uu uu
r r r
+
u
1( )
2 = +u u z r r
zz rr
1u u u u= , = , = +
z r r r
1 1 1 1 1( ) ( ) ( )2 2 2
r z r z r z rz
u u uu u u u= + , = + , = +
r r r r z z r
1 1r zr
uu uu
r r r z
= + + +
u
1 1( ) ( ) ( )r zu = +e e e
z r z ru u uu u u u+r z z r r r r
+
2 2 2
2
2 2 2 2
1 1f f f ff = f = + + +
r rr r z
Some formulas in cylindrical coordinate P-28
P 29
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011
2=++ u
r
u
r
u rrr
1
x y r r
= + = + x y re e e e
2 1 1( ) ( )u uu ur r r r
= = + + r re e e e
2 22
2 2( ) ( ) ( )
u uu u u
x y x y x y
= = + + = +
x y x ye e e e
=
re er
=
ee
d
2
2
2 2
2 2
2 2
2 2 2
2
2 2
1( )
1 1 1 1 1 1 1( ) [ ( ) ( ) ]
1 1 1 1[ ]
1 1
u u u
r r r r
u u u u u ur r r r r r r r r r
u u u u
r r r r r r
u u
r r r
+ =
+ = + + +
= + +
= +
r r
rr r
r r
e e e
eee e e e e e
e e e e e
re
e
Laplace equation in 2-D polar form
1 =r re e 0 =re e 1 =e e
Example P-12
Will be useful later!
P-29
R i H b li f ti P 30
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Review on Hyperbolic functions
2sinh
xx eex
=
1csch
sinhx
x=
cosh2
x xe ex
+=
1sech
coshx
x=
x
xx
cosh
sinhtanh =
x
xx
sinh
coshcoth =
xx sinh)sinh( =
cosh( ) coshx x =
1sinhcosh 22 = xx2 2sinh cosh 1x x=
2 2cos sin 1x x+ =
tanh x
sinh
cosh x
P-30
i h h i P 31
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sinhx, coshx versus sinx, cosx P-31
Lambert (17281777)
V Riccati (17071775)
P-32
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sinhx, coshx versus sinx, cosx
hyperbola circle
Hyperbolicfunctions
Circular
functionsResemblance
2 21 tanh sechx x =
yxyxyx sinhcoshcoshsinh)sinh( +=+
yxyxyx sinhsinhcoshcosh)cosh( +=+
xxdxd cosh)(sinh =
xxdx
dsinh)(cosh =
2(tanh ) sechd x xdx
=
(sech ) sech tanhd x x xdx =
2(coth ) cschd x xdx
=
(csch ) csch cothd
x xdx
=
1
2
1(sinh )1
dx
dx x
=+
1
2
1(cosh )
1
dx
dx x
=
1
2
1(tanh )
1
dx
dx x
=
1
2
1(sech )
1
dx
dx x x
=
1
2
1
(coth ) 1
d
xdx x
=
1
2
1(csch )
1
dx
dx x x
= +
P-32
2sinh
xx eex
= cosh
2
x xe ex
+=
P-33
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2
sinh cosh
cosh sinh
sech tanh
udu u C
udu u C
udu u C
= +
= +
= +
2
csch cot csch
sech tanh sech
csch coth
u udu u C
u udu u C
udu u C
= +
= +
= +
1
2 2
12 2
1 2
2 21 2
sinh
cosh
1tanh
1coth
2
2
if u
if u
du uC
aa u
du u Cau a
uC a
du a a
ua uC aa a
= ++
= +
+
1
2 2
1
2 2
1sec 0
1
csc 0
du u
h C u aa au a u
du u
h C ua au u a
= +