Mathematical Tools for Physics

2—Infinite Series 61

2.30 You know the power series representation for the exponential function, but now apply it in a slightly different
context. Write out the power series for the exponential, but with an argument that is a differential operator. The
letterhrepresents some fixed number; interpret the square ofd/dxasd^2 /dx^2 and find

eh

d

dxf(x)

Interpret the terms of the series and show that the value of this isf(x+h).

2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The
Doppler effect for light, including relativistic effects is different still. Show that for low speeds they are all about
the same.

f′=f

v−vo

v

f′=f

v

v+vs

f′=f

√

1 −v/c

1 +v/c

The symbols have various meanings: vis the speed of sound in the first two, with the other terms being the
velocity of the observer and the velocity of the source. In the third equationcis the speed of light andvis the
velocity of the observer. (And no,1 = 1isn’t good enough.)

2.32 In the equation ( 21 ) for the light diffracted through a narrow slit, the width of the central maximum is
dictated by the angle at the first dark region. How does this angle vary as you vary the width of the slit,a? What
is this angle ifa= 0. 1 mm andλ= 700nm? And how wide will the central peak be on a wall 5 meters from the
slit? Take this width to be the distance between the first dark regions on either side of the center.

2.33 An object is a distancedbelow the surface of a medium with index of refractionn. (For example, water.)
When viewed from directly above the object in air (i.e. use small angle approximation), the object appears to
be a distance below the surface given by (maybe) one of the following expressions. Show why most of these
expressions are implausible; that is, give reasons for eliminating the wrong ones.

(1)

d

√

1 +n^2

n

(2)

dn

√

1 +n^2

(3)nd (4)

d

n

(5)

dn^2

√

1 +n^2