1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

436 Appendix: Mathematical References

Hyperbolic Functions

cosh(A)=

1

2

(

eA+e−A

)

, sinh(A)=

1

2

(

eA−e−A

)

dcosh(u)=sinh(u)du, dsinh(u)=cosh(u)du

sinh(A±B)=sinh(A)cosh(B)±cosh(A)sinh(B)

cosh(A±B)=cosh(A)cosh(B)±sinh(A)sinh(B)

sinh(A)+sinh(B)=2sinh

(A+B

2

)

cosh

(A−B

2

)

sinh(A)−sinh(B)=2cosh

(A+B

2

)

sinh

(A−B

2

)

cosh(A)+cosh(B)=2cosh

(

A+B

2

)

cosh

(

A−B

2

)

cosh(A)−cosh(B)=2sinh

(

A+B

2

)

sinh

(

A−B

2

)

sinh(A)sinh(B)=

1

2

(

cosh(A+B)−cosh(A−B)

)

sinh(A)cosh(B)=^12

(

sinh(A+B)+sinh(A−B)

)

cosh(A)cosh(B)=^1

2

(

sinh(A+B)+cosh(A−B)

)

cosh^2 (A)−sinh^2 (A)= 1 , 1 −tanh^2 (A)=sech^2 (A)

Calculus

1.Derivative of a product

(uv)′=u′v+uv′

(uv)′′=u′′v+ 2 u′v′+uv′′

(uv)(n)=u(n)v+

(n

1

)

u(n−^1 )v′+···+

( n

n− 1

)

uv(n−^1 )+uv(n)

In this formula,

(n

k

)

=(n−nk!)!k!is a binomial coefficient.

2.Rules of integration

a.

∫b

a

(

c 1 f 1 (x)+c 2 f 2 (x)

)

dx=c 1

∫b

a

f 1 (x)dx+c 2

∫b

a

f 2 (x)dx