An in-depth and concise summary of engineering mathematics procedures. Covers concepts such as integration techniques, complex numbers, limits, hyperbolic trigonometry, inverse trigonometry, linear systems, matrices, and partial fractions, among others
We define the compless bi bi
conjugate of number Di If then-E R
=
a -= -
any as
-
-= a + - =
+
a -
.
,
, Geometric representation of compless numbers :
Im
X
Re(-) By (m) -)
iy determined by ordered (C
y) where
:
-= x an pair x
+ = =
,
,
·
Re
it i le me
relationship
...
x
-
=
-
rcOSO
rcoso
between
+
3
irsing
y
= using
=
-=xc+yi3
r(cos6+isinG)
the
length angle' representation is
Standard form of - : - = x +
yi where x =
ReC -1 3y = 1mC-) with cartesian coordinates (sc , y)
,
.
Polar form of --=r(cosotising) where roso
=
Re(-)3 usino =
1mC-) with polar coordinates Cr ol ,
,
1-1
Modulus x
yi x yz
=
:
r
+
of - = + =
- -
=
Argument of - :
arg)-)= arctan ? =
& + 2nk
,
KE .
If Re(-) Co
,
add subtract i to from a The Principle argument of -
is the
unique argument Arg(-) =
8
,
where 0 =(-4in) .
Note :
for two complex numbers -=r(cosx+ isinx) 3 w =
s(cos +
isin) ,
we have that
-w
=
rs(cos(x +
3) +
isin(x +
B)) 3 w =
(cos(a -
B) +
isin(a-5)) .
Sub-note for any -
,
wD ,
-W = -
-w > i =
i
arg)-w)
=
args -) +
arg(w) S arg(n) =
a w!
I heorem De Moirre's formula ne complex r(cosO isino) ,
:
for any ↳ number -= +
,
·
(r(coso+isina))" =
r(cos(no) + isin(nal)
cor) .
-=-1 arg(-r) =
narg(-)
Theorem
:
for a
positive integer n b a non-zero complex number -
let o be an
argument of -
Then all the nith roots of -
are
given by the formula :
isn( ** )
*
-(cos(0 )
+
n
for 90 n-13
·
K =
+
1 2
=
-
, , . . .,
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