Formula Sheet - Final Exam
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1
2
3Math 257-316 PDE
Formula sheet - final exam
1
m
t
m(m
1)(m
2)
π. In general,
(m + 1) =
t
e
dt for real
2
0
2
m
0. Special case: J
(x) =
sin x.
1/2
πx
Power series and analytic functions:
(t can be a complex number)
Trigonometric identities
A function f (x) is analytic at a point x
if it has a power series expansion
0
2
2
sin(α
β) = sin α cos β
sin β cos α
sin
t + cos
t = 1
in powers of x
x
with radius R > 0,
0
2
1
cos(α
β) = cos α cos β
sin β sin α.
sin
t =
(1
cos(2t))
2
1
1
2
sin α cos β =
[sin(α
β) + sin(α + β)]
cos
t =
(1 + cos(2t))
n
f (x) =
a
(x
x
)
,
x
x
< R, R > 0.
2
2
n
0
0
n=0
1
cos α cos β =
[cos(α
β) + cos(α + β)]
sin(2t) = 2 sin t cos t
2
1
2
2
sin α sin β =
[cos(α
β)
cos(α + β)], cos(2t) = 2 cos
t
1 = 1
2 sin
t
The sum and product of analytic functions are still analytic. Their quotient is
2
x sin ωx
cos ωx
x cos ωx
sin ωx
also analytic if the denominator is nonzero. The new function has a radius of
x cos ωx dx =
+
,
x sin ωx dx =
+
2
2
ω
ω
ω
ω
convergence no smaller than the minimum of those of the original functions.
2
x
cos ωx
2x sin ωx
2 cos ωx
2
x
sin ωx dx =
+
+
1
m
t
1
k
=
t
,
e
=
t
.
2
3
ω
ω
ω
1 t
m=0
k=0
k!
Fourier, sine and cosine series
m
m
it
it
it
it
( 1)
( 1)
e
+e
e
e
2m
2m+1
cos t =
t
=
,
sin t =
t
=
.
m=0
m=0
(2m)!
2
(2m+1)!
2i
Suppose f (x) is a function defined in [ L, L]. Its Fourier series is
t
t
t
t
1
e
+e
1
e
e
2m
2m+1
cosh t =
t
=
,
sinh t =
t
=
.
m=0
m=0
(2m)!
2
(2m+1)!
2
nπx
nπx
F f (x) = a
+
a
cos(
) + b
sin(
)
ix
0
n
n
e
= cos x + i sin x,
cos t = cosh(it),
i sin t = sinh(it).
L
L
n=1
k
1
( 1)
k
1
k
ln(1 + t) =
t
,
ln(1
t) =
t
.
L
k=1
k=1
k
k
1
where a
=
f (x) dx,
0
2L
L
L
L
Basic linear ODE’s with real coefficients
1
nπx
1
nπx
a
=
f (x) cos(
) dx,
b
=
f (x) sin(
) dx (n
1).
n
n
L
L
L
L
L
L
constant coefficients
Euler eq
2
ODE
ay + by + cy = 0
ax
y + bxy + cy = 0
F f (x) is defined for all real x and is a 2L-periodic function.
2
indicial eq.
ar
+ br + c = 0
ar(r
1) + br + c = 0
Theorem (Pointwise convergence)
If f (x) and f (x) are piecewise con-
r
x
r
x
r
r
1
r
= r
real
y = Ae
+ Be
y = Ax
+ Bx
1
2
1
2
tinuous, then F f (x) converges for every x to
[f (x ) + f (x+)].
1
2
2
rx
rx
r
r
r
= r
= r
y = Ae
+ Bxe
y = Ax
+ Bx
ln x
Theorem (Square norm convergence)
If f (x) is square integrable,
1
2
λx
λ
L
r = λ
iµ
e
[A cos(µx) + B sin(µx)]
x
[A cos(µ ln x ) + B sin(µ ln x )]
2
i.e.,
f (x)
dx <
. Then F f (x) converge to f (x) in square norm, i.e.
L
L
2
f (x)
F
f (x)
dx
0 as n
, here F
f (x) denotes the partial sum
n
n
Bessel equations of order p
(useful in polar coordinates)
L
of F f (x). Moreover, (Parseval’s indentity)
Eq:
L
2
2
2
1
x
y + xy + (x
p
)y = 0.
2
2
2
2
f (x)
dx = 2 a
+
a
+ b
.
0
n
n
L
L
Bessel function of first kind of order p,
n=1
L
1
k
2k+p
( 1)
For f (x) defined in [0, L], its cosine and sine series are (a
=
f (x) dx)
x
J
(x) =
.
0
L
0
p
k=0
k! (k+p+1)
2
L
nπx
2
nπx
Here
is the Gamma function, which generalizes the factorial function. If
Cf (x) = a
+
a
cos(
),
a
=
f (x) cos(
) dx,
0
n
n
L
L
L
m is an integer,
(m + 1) = m!; If m
1/2 is an integer,
(m + 1) =
0
n=1
1
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