to PHY226 Mathematical Methods for Physics and Astronomy

1
to PHY226Mathematical Methods for Physics
and Astronomy
Welcome!

• Purpose of the course
• To provide the further mathematical knowledge and
  skills needed to fully complete future physics
  courses
• If you missed Monday pick up notes now

Phil Lightfoot, E47, (24533) p.k.lightfoot_at_shef.ac
.uk
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IQ test time!!!!
If an even function is defined by f(x)f(-x) and
an odd function is defined by f(x) -f(-x), are
the following even or odd
are these even, odd, or just a mess.?
(a) cos2(x) (b) sin2x (c) sin(x)cos(x) (d) ex
(e) e-x (f) 0.5(ex e-x)
How does this observation allow you to short cut
integration across the origin for any odd or even
function?
3
IQ test time!!!!
If an even function is defined by f(x)f(-x) and
an odd function is defined by f(x) -f(-x), are
the following even or odd
(a) cos2(x)
Cos(x)
Cos2(x)
So an even function x an even function gives an
even function
4
IQ test time!!!!
If an even function is defined by f(x)f(-x) and
an odd function is defined by f(x) -f(-x), are
the following even or odd
(b) sin2(x)
Sin(x)
Sin2(x)
So an odd function x an odd function gives an
even function
5
IQ test time!!!!
If an even function is defined by f(x)f(-x) and
an odd function is defined by f(x) -f(-x), are
the following even or odd
(c) sin(x)cos(x)
sin(x)
sin(x)cos(x)
cos(x)
So an odd function x an even function gives an
odd function
6
IQ test time!!!!
If an even function is defined by f(x)f(-x) and
an odd function is defined by f(x) -f(-x), are
the following even or odd
(d) ex
(e) e-x
exp(x)
exp(-x)
(f) 0.5(ex e-x)
Single exponentials are neither but adding gives
an even function (coshx) and subtracting gives an
odd function.
7
IQ test time!!!!
So even x even even even x odd odd
odd x odd even
How does this observation allow you to short cut
integration across the origin for any odd or even
function?
8
Complex numbers
Argand diagram
Cartesian a ib
Imaginary
r
b
Real
q
a
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Complex numbers
Argand diagram
z a ib
Imaginary
r
b
Real
q
a
z is the complex conjugate
z a - ib
zz(aib)(a-ib)a2b2r2
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Complex numbers
Argand diagram
Cartesian a ib
Imaginary
r
b
Real
q
a
Polar
so
where
11
Complex numbers
Polar
But remember earlier we showed that
and
and
So
12
Complex numbers
So whats the point of complex numbers?
SHM / LHO / Schrodinger / Diffusion of gas /
Diffusion of heat / all of optics / standing
waves /
All these rely heavily on a mixture of
trigonometry and calculus
Adding Subtracting
Replacing all the trig with exp makes the
calculus trivial!!!!!
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Working with complex numbers
Add / subtract
Multiply / divide
Powers
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Working with complex numbers
Roots
Example Step 1 write down z in polars with
the 2pp bit added on to the argument. Step 2
say how many values of p youll need (as many as
n) and write out the rooted expression ..
Step 3 Work it out for each value of p.
If what is z½?
here n 2 so Ill need 2 values of p p 0 and
p 1.
p 0
p 1