Higher Engineering Mathematics, Sixth Edition

Essential formulae 663

Complex Numbers

z=a+jb=r(cosθ+jsinθ)=r∠θ=rejθ where

j^2 =− 1

Modulusr=|z|=

√

(a^2 +b^2 )

Argumentθ=argz=tan−^1

b

a

Addition:(a+jb)+(c+jd)=(a+c)+j(b+d)

Subtraction:(a+jb)−(c+jd)=(a−c)+j(b−d)

Complex equations:Ifm+jn=p+jqthenm=p
andn=q

Multiplication:z 1 z 2 =r 1 r 2 ∠(θ 1 +θ 2 )

Division:
z 1
z 2

=

r 1

r 2

∠(θ 1 −θ 2 )

De Moivre’s theorem:

[r∠θ]n=rn∠nθ=rn(cosnθ+jsinnθ)=rejθ

Matrices and Determinants

Matrices:

IfA=

(

ab

cd

)

and B=

(

ef

gh

)

then

A+B=

(

a+eb+f

c+gd+h

)

A−B=

(

a−eb−f

c−gd−h

)

A×B=

(

ae+bg af+bh

ce+dg cf+dh

)

A−^1 =

1

ad−bc

(

d −b

−ca

)

If A=

⎛

⎜

⎝

a 1 b 1 c 1

a 2 b 2 c 2

a 3 b 3 c 3

⎞

⎟

⎠ then A−^1 =

BT

|A|

where

BT=transpose of cofactors of matrix A

Determinants:

∣

∣

∣

∣

ab

cd

∣

∣

∣

∣=ad−bc

∣ ∣ ∣ ∣ ∣ ∣

a 1 b 1 c 1

a 2 b 2 c 2

a 3 b 3 c 3

∣ ∣ ∣ ∣ ∣ ∣

=a 1

∣

∣

∣

∣

b 2 c 2

b 3 c 3

∣

∣

∣

∣−b^1

∣

∣

∣

∣

a 2 c 2

a 3 c 3

∣

∣

∣

∣

+c 1

∣

∣

∣

∣

a 2 b 2

a 3 b 3

∣

∣

∣

∣

Differential Calculus

Standard derivatives:

yorf(x)

dy

dx

orf′(x)

axn anxn−^1

sinax acosax

cosax −asinax

tanax asec^2 ax

secax asecaxtanax

cosecax −acosecaxcotax

cotax −acosec^2 ax

eax aeax

lnax

1

x

sinhax acoshax

coshax asinhax

tanhax asech^2 ax

sechax −asechaxtanhax

cosechax −acosechaxcothax

cothax −acosech^2 ax

sin−^1

x

a

1

√

a^2 −x^2

sin−^1 f(x)

f′(x)

√

1 −[f(x)]^2

cos−^1

x

a

− 1

√

a^2 −x^2