### The Maxima on-line user's manual

Algebra Calculator

#### Hankel_1

Function: hankel_1 (<v>, <z>) The Hankel function of the first kind of order v and argument z (A&S 9.1.3). `hankel_1` is defined as

bessel_j(v,z) + %i * bessel_y(v,z)

Maxima evaluates `hankel_1` numerically for a real order v and complex argument z in float precision. The numerical evaluation in bigfloat precision and for a complex order v is not supported.

When `besselexpand` is `true`, `hankel_1` is expanded in terms of elementary functions when the order v is half of an odd integer. See `besselexpand`.

Maxima knows the derivative of `hankel_1` wrt the argument z.

Examples:

Numerical evaluation:

```          (%i1) hankel_1(1,0.5);
(%o1)              .2422684576748738 - 1.471472392670243 %i
(%i2) hankel_1(1,0.5+%i);
(%o2)             - .2558287994862166 %i - 0.239575601883016```

A complex order v is not supported. Maxima returns a noun form:

```          (%i3) hankel_1(%i,0.5+%i);
(%o3)                       hankel_1(%i, %i + 0.5)```

Expansion of `hankel_1` when `besselexpand` is `true`:

```          (%i4) hankel_1(1/2,z),besselexpand:true;
sqrt(2) sin(z) - sqrt(2) %i cos(z)
(%o4)                 ----------------------------------
sqrt(%pi) sqrt(z)```

Derivative of `hankel_1` wrt the argument z. The derivative wrt the order v is not supported. Maxima returns a noun form:

```          (%i5) diff(hankel_1(v,z),z);
hankel_1(v - 1, z) - hankel_1(v + 1, z)
(%o5)               ---------------------------------------
2
(%i6) diff(hankel_1(v,z),v);
d
(%o6)                        -- (hankel_1(v, z))
dv```

```(%o1)                                true
(%i2) ```

### Related Examples

##### hankel_1

hankel_1(2/3,0.5);

Calculate

##### hankel_1

hankel_1(2/3,0.5);

hankel_1(2/5,0.5);

Calculate

? hankel_1;

Calculate

hankel_1(1,0.5);

Calculate

##### hankel_1

hankel_1(2/3,0.5);

Calculate

##### hankel_1

hankel_1(2/3,0.5);

hankel_1(2/5,0.5);

Calculate

? hankel_1;

Calculate

hankel_1(1,0.5);

Calculate