engineering mathematics formula (1)
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1 Engineering Mathematics Formula
Ali Jabari Moghadam
( ) 01
2 2 2 2cos sin , cosn n n
n
n n nf x a a x b x a f x x dx
T T T T
π π π∞
=
= + + =
∑
( ){ } ( ) (
( ){ } ( ) (1
exp2
1exp
kF f x f x i x dx
F F F i x dk
π
ω ω ω ω
∞
−∞∞
−
−∞
/ = ±
/ =
∫
∫
( ){ } ( ) ( )1/ , 0F f ax F a a
aω/ = >
( ){ } ( ) (1, , , exp
2x F f x t F t f x t i x dxω ω
π
∞
−∞
/ = = −∫
( ) ( ) ( ), ,n
n
x nF f x t i F t
xω ω
∂ / = ∂
x
( ) ( )0
2cosCF f x xdxω ω
π
∞
= ∫
( ) ( )0
2cosCf x F xdω ω ω
π
∞
= ∫
( ){ } ( ){ } ( )20C SF f x F f x fω
π′/ = / −
( ){ } ( ){ } ( )2 20C CF f x F f x fω
π′′ ′/ = − / −
F f x F f x f/ = − / +
( )
( )
( )
0
1
2 2 2 2cos sin , cos
2 2sin
c T
c
c T
n n n
c
c T
n
c
a f x dxT
n n nf x a a x b x a f x x dx
T T T T
nb f x x dx
T T
π π π
π
+
+
+
=
= + + =
=
∫
∫
∫
( ) ( ) 2lim 0nn
f x S x dxπ
π→∞
−
− = ∫
( )2 2 20
1
12 r r
r
a a b f x dxπ
ππ
∞
= −
+ + = ∑ ∫
)
)exp
F f x f x i x dx
F F F i x d
ω
ω ω ω ω
/ = ±
∓
( ) ( ){ } ( )nnF f x i Fω ω/ =
( ){ }n
n nn
dF x f x i F
dω/ =
( ) ( ){ } ( )m
m nnm ndF x f x i F
dω+/ =
( ){ } ( ) ( )expF f x a i a Fω ω/ − = − ( ){expF i x f x Fλ ω λ/ = −
) ( ), , , expF f x t F t f x t i x dxω ω/ = = −
( ) ( ){ }1 1, , , exp
2xf x t F F t F t i x dω ω ω ω
π
∞−
−∞
= / = ∫
( ){ } ( ), ,n
n nx nF x f x t i F tω
ω∂
/ = ∂
( ), ,n m
m m n nx n mF x f x t i F t
x
∂ ∂ / = ∂ ∂
F f x xdx
f x F xdω ω ω
( ) ( )0
2sinSF f x xdxω ω
π
∞
= ∫
( ) ( )0
2sinSf x F xdω ω ω
π
∞
= ∫
( ){ } ( ){ }S CF f x F f xω′/ = − /
( ){ } ( ){ } ( )2 20S SF f x F f x fω ω
π′′/ = − / +
f x a a x b x a f x x dx
b f x x dx
2lim 0f x S x dx− =
( ) 2a a b f x dx ∫
( )nF f x i Fω ω
( )nF x f x i F ω
( )m
m nm
dF x f x i Fω ω
ω
( )} ( )F i x f x Fλ ω λ/ = −
( ) ( ), , , expf x t F F t F t i x dω ω ω ω∞
−∞∫
) ( ), ,n m
m m n nn m
F x f x t i F tω ωω
+ ∂ ∂ / = ∂ ∂
sinF f x xdxω ω
sinf x F xdω ω ω
2 Engineering Mathematics Formula
Ali Jabari Moghadam
( ) ( ){ } ( ){1cos
2C C CF ax f x F a F aω ω/ = + + −
( ) ( ){ } ( ){1cos
2S S SF ax f x F a F aω ω/ = + + −
( ){ } ( )1, 0C CF f ax F a a
aω/ = >
( ){ } ( ) (0
2, , , cosx C CF f x y F y f x y xdxω ω
π
∞
/ = = ∫
( ){ } ( ), , 0,x C SF f x t F t f tω ω′/ = −
( ){ } ( )2, , 0,x C SF f x t F t f tω ω′′ ′/ = − −
( ) (, ,n nC
x C n n
f x y d F yF
y dy
ω ∂ / = ∂
( )x
x
x
AeF
or
xCF
FFµ
µµ
+=
+=⇒=−′′
)cosh(
0)(
2
( )}C C CF ax f x F a F aω ω/ = + + −
( ) ( ){ } ({1sin
2C S SF ax f x F a F aω ω/ = + + −
( )}S S SF ax f x F a F aω ω/ = + + −
( ) ( ){ } ({1sin
2S C CF ax f x F a F aω ω/ = − − +
( ), 0F f ax F a a/ = >
( ){ } ( )1S SF f ax F a a
aω/ = >
( ), , , cosF f x y F y f x y xdxω ω∫
( ){ } ( ) 2, , , siny S SF f x y F x f x y ydyω ω
π/ = =
( )2, , 0,F f x t F t f t
π
( ){ }, ,x S CF f x t F tω ω′/ = −
( )2, , 0,F f x t F t f t
π′′ ′/ = − −
( ){ } (2, , 0,x S SF f x t F t f tω ω ω′′ ′/ = − +
), ,n n
f x y d F y
y dy
ω
xBe
xD
µ
µ
−+
+ )sinh(
( )02 AFFF xµ =⇒=+′′
) ( )}C S SF ax f x F a F aω ω/ = + + −
) ( )}S C CF ax f x F a F aω ω/ = − − +
) ( ), 0F f ax F a a/ = >
( )0
2, , , sinF f x y F x f x y ydyω ω
π
∞
∫
( ), ,x S CF f x t F tω ω
) ( )2, , 0,F f x t F t f tω ω ω
π′′ ′/ = − +
)sin()cos( xBxA µµ +
3 Engineering Mathematics Formula
Ali Jabari Moghadam
( ) ( )∑∞
=−=
00
n
n
n zzazf
----------------------------------
---------------------
سري تيلور
---------------------
4 Engineering Mathematics Formula
Ali Jabari Moghadam
( ) ( )( )
((0
2 Resy
P x P zf x dx dx i
Q x Q zπ
+∞ +∞
>−∞ −∞
= = ∑∫ ∫
( ) ( ) ( )0 0
1
01
1Res lim
1 !
NN
Nz z z z
df z z z f z
N dz
−
−= → = − −
سري لورنت
( ) ( ) ( )
( ) ( ) ( )0
cos 2 Im Res
sin 2 Re Res
y
y
f x ax dx f z e
f x ax dx f z e
π
π
+∞
>−∞
+∞
>−∞
= −
=
∑∫
∑∫
( )( )
P x P z
Q x Q z
( )Nf z z z f z
( )
( )
0
cos 2 Im Res
sin 2 Re Res
iaz
y
iaz
f x ax dx f z e
f x ax dx f z e
>
∑
∑